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Existence globale pour des systemes de reaction-diffusion aveccontrole de la masse totale
Global existence for reaction-diffusion systems with a control of thetotal mass
Michel Pierre
Ecole Normale Superieure de Renneset Institut de Recherche Mathematique de Rennes (IRMAR)
Bretagne, France
Journees EDP et Modeles en Bio-Mathematique,organisees par le GE2MI, les 15-16 decembre 2021
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I About reaction-diffusion systems with two main properties,very frequent in applications :
I (P) : Positivity of the solution is preserved for all time
I (M) : Conservation or dissipation of the total mass holds
I or (M’) : at least a control of the total mass for all time.
I Often in Applications: (Bio-)Chemical reactions, populationdynamics, Lotka-Volterra,...
I (P)+(M’) imply that the L1-norm of the solution iscontrolled for all time; thus it cannot blow up in finite time.
I But this is not sufficient to imply global existence for all timeof solutions.
I And we will mainly discuss this question of global existence,together with some results on the asymptotic behavior forlarge time.
Goal of the lectures
I A main question will be:Blow up or not blow up in finite time???
I It turns out to be a difficult question, still in progress. We willfirst recall the main positive and negative results(L∞ − Lp − L1 − L2 techniques...).
I A major progress has been made recently in the understandingof global existence of classical solutions, for systems withquadratic reaction-terms. This is the case for lots of systemsarising in chemistry and biology.
I I will describe this recent result. It is very impressive andbased on regularity properties for the standard linear heatequation and for linear diffusion equations with (only)bounded coefficients and in non-divergence form. Theseproperties are of independent interest.
I And I will discuss the asymptotic behavior as t → +∞ forsome of these systems ( some techniques of general interest ).
Goal of the lectures
I A main question will be:Blow up or not blow up in finite time???
I It turns out to be a difficult question, still in progress. We willfirst recall the main positive and negative results(L∞ − Lp − L1 − L2 techniques...).
I A major progress has been made recently in the understandingof global existence of classical solutions, for systems withquadratic reaction-terms. This is the case for lots of systemsarising in chemistry and biology.
I I will describe this recent result. It is very impressive andbased on regularity properties for the standard linear heatequation and for linear diffusion equations with (only)bounded coefficients and in non-divergence form. Theseproperties are of independent interest.
I And I will discuss the asymptotic behavior as t → +∞ forsome of these systems ( some techniques of general interest ).
Goal of the lectures
I A main question will be:Blow up or not blow up in finite time???
I It turns out to be a difficult question, still in progress. We willfirst recall the main positive and negative results(L∞ − Lp − L1 − L2 techniques...).
I A major progress has been made recently in the understandingof global existence of classical solutions, for systems withquadratic reaction-terms. This is the case for lots of systemsarising in chemistry and biology.
I I will describe this recent result. It is very impressive andbased on regularity properties for the standard linear heatequation and for linear diffusion equations with (only)bounded coefficients and in non-divergence form. Theseproperties are of independent interest.
I And I will discuss the asymptotic behavior as t → +∞ forsome of these systems ( some techniques of general interest ).
Goal of the lectures
I A main question will be:Blow up or not blow up in finite time???
I It turns out to be a difficult question, still in progress. We willfirst recall the main positive and negative results(L∞ − Lp − L1 − L2 techniques...).
I A major progress has been made recently in the understandingof global existence of classical solutions, for systems withquadratic reaction-terms. This is the case for lots of systemsarising in chemistry and biology.
I I will describe this recent result. It is very impressive andbased on regularity properties for the standard linear heatequation and for linear diffusion equations with (only)bounded coefficients and in non-divergence form. Theseproperties are of independent interest.
I And I will discuss the asymptotic behavior as t → +∞ forsome of these systems ( some techniques of general interest ).
Goal of the lectures
I A main question will be:Blow up or not blow up in finite time???
I It turns out to be a difficult question, still in progress. We willfirst recall the main positive and negative results(L∞ − Lp − L1 − L2 techniques...).
I A major progress has been made recently in the understandingof global existence of classical solutions, for systems withquadratic reaction-terms. This is the case for lots of systemsarising in chemistry and biology.
I I will describe this recent result. It is very impressive andbased on regularity properties for the standard linear heatequation and for linear diffusion equations with (only)bounded coefficients and in non-divergence form. Theseproperties are of independent interest.
I And I will discuss the asymptotic behavior as t → +∞ forsome of these systems ( some techniques of general interest ).
Outline of the lectures
I A) Recalling the main results on the (P)+(M)systems: L∞ − Lp − L1 − L2-techniques.
I B) The recent result of existence of globalclassical solutions for quadratic systems; themain ideas behind (includes Lotka-Volterra andothers).
I C) Some results on asymptotic behaviors forlarge time.
Outline of the lectures
I A) Recalling the main results on the (P)+(M)systems: L∞ − Lp − L1 − L2-techniques.
I B) The recent result of existence of globalclassical solutions for quadratic systems; themain ideas behind (includes Lotka-Volterra andothers).
I C) Some results on asymptotic behaviors forlarge time.
Outline of the lectures
I A) Recalling the main results on the (P)+(M)systems: L∞ − Lp − L1 − L2-techniques.
I B) The recent result of existence of globalclassical solutions for quadratic systems; themain ideas behind (includes Lotka-Volterra andothers).
I C) Some results on asymptotic behaviors forlarge time.
Outline of the part A): the main results on (P)+(M)systems
I Description of the general RD-systems with their easyproperties (L∞-local existence).-EXERCICES-
I Some finite time blow up examples with anysuper-quadratic growth
I Global existence of classical solutions for ”triangular”systems or for closed diffusion coefficients: Lp-dualitytechnique -EXERCICES-
I Global existence of weak ”L1”-solutions
I A main L2-estimate
Outline of the part A): the main results on (P)+(M)systems
I Description of the general RD-systems with their easyproperties (L∞-local existence).-EXERCICES-
I Some finite time blow up examples with anysuper-quadratic growth
I Global existence of classical solutions for ”triangular”systems or for closed diffusion coefficients: Lp-dualitytechnique -EXERCICES-
I Global existence of weak ”L1”-solutions
I A main L2-estimate
Outline of the part A): the main results on (P)+(M)systems
I Description of the general RD-systems with their easyproperties (L∞-local existence).-EXERCICES-
I Some finite time blow up examples with anysuper-quadratic growth
I Global existence of classical solutions for ”triangular”systems or for closed diffusion coefficients: Lp-dualitytechnique -EXERCICES-
I Global existence of weak ”L1”-solutions
I A main L2-estimate
Outline of the part A): the main results on (P)+(M)systems
I Description of the general RD-systems with their easyproperties (L∞-local existence).-EXERCICES-
I Some finite time blow up examples with anysuper-quadratic growth
I Global existence of classical solutions for ”triangular”systems or for closed diffusion coefficients: Lp-dualitytechnique -EXERCICES-
I Global existence of weak ”L1”-solutions
I A main L2-estimate
Outline of the part A): the main results on (P)+(M)systems
I Description of the general RD-systems with their easyproperties (L∞-local existence).-EXERCICES-
I Some finite time blow up examples with anysuper-quadratic growth
I Global existence of classical solutions for ”triangular”systems or for closed diffusion coefficients: Lp-dualitytechnique -EXERCICES-
I Global existence of weak ”L1”-solutions
I A main L2-estimate
Description of the general family of systemsI For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× ∂Ω
Ω ⊂ IRN open bounded with regular boundary,
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi(u1, u2, ..., um) in QT ,∂νui = 0 [ for the lecture], on ΣT ,ui(0, ·) = u0
i (·) ≥ 0.
where ∆ =∑N
k=1 ∂kk , ui = ui (t, x), (t, x) ∈ QT the unknown,
di ∈ (0,+∞), fi : [0,∞)m → IR locally Lipschitz continuous (C 1 for
instance).
I Thanks to the Neumann homogeneous boundarycondition, note the possibility of solutions independent ofx , i.e. solution of the O.D.E
(ODE )
∀i = 1, ...,m,∂tui = fi(u1, ..., um),ui(0) = u0
i ∈ [0,+∞).
Description of the general family of systemsI For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× ∂Ω
Ω ⊂ IRN open bounded with regular boundary,
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi(u1, u2, ..., um) in QT ,∂νui = 0 [ for the lecture], on ΣT ,ui(0, ·) = u0
i (·) ≥ 0.
where ∆ =∑N
k=1 ∂kk , ui = ui (t, x), (t, x) ∈ QT the unknown,
di ∈ (0,+∞), fi : [0,∞)m → IR locally Lipschitz continuous (C 1 for
instance).
I Thanks to the Neumann homogeneous boundarycondition, note the possibility of solutions independent ofx , i.e. solution of the O.D.E
(ODE )
∀i = 1, ...,m,∂tui = fi(u1, ..., um),ui(0) = u0
i ∈ [0,+∞).
Description of the general family of systems
I For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× Ω
Ω ⊂ IRN open bounded with regular boundary,
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi(u1, u2, ..., um) in QT ,∂νui = 0 [ for the lecture], on ΣT ,ui(0, ·) = u0
i (·) ≥ 0.
ui = ui (t, x), di ∈ (0,+∞), fi : [0,∞)m → IR regular
I and f = (f1, ..., fm) satisfies the two main followingproperties:
I (P): Positivity is preserved for all time for the solutions of(ST )
I (M):∑
1≤i≤m fi(r) ≤ 0, ∀ r = (r1, ..., rm) ∈ [0,+∞)m.
Description of the general family of systems
I For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× Ω
Ω ⊂ IRN open bounded with regular boundary,
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi(u1, u2, ..., um) in QT ,∂νui = 0 [ for the lecture], on ΣT ,ui(0, ·) = u0
i (·) ≥ 0.
ui = ui (t, x), di ∈ (0,+∞), fi : [0,∞)m → IR regular
I and f = (f1, ..., fm) satisfies the two main followingproperties:
I (P): Positivity is preserved for all time for the solutions of(ST )
I (M):∑
1≤i≤m fi(r) ≤ 0, ∀ r = (r1, ..., rm) ∈ [0,+∞)m.
Description of the general family of systems
I For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× Ω
Ω ⊂ IRN open bounded with regular boundary,
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi(u1, u2, ..., um) in QT ,∂νui = 0 [ for the lecture], on ΣT ,ui(0, ·) = u0
i (·) ≥ 0.
ui = ui (t, x), di ∈ (0,+∞), fi : [0,∞)m → IR regular
I and f = (f1, ..., fm) satisfies the two main followingproperties:
I (P): Positivity is preserved for all time for the solutions of(ST )
I (M):∑
1≤i≤m fi(r) ≤ 0, ∀ r = (r1, ..., rm) ∈ [0,+∞)m.
I About the condition (P):
(E)
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity(quasipositivity of f ): ∀i = 1, ...,m∀r = (r1, ..., rm) ∈ [0,∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
6r2
-r1
f1(0, r2)≥ 0
f2(0, r2)
AAAKf1(r1, 0)
f2(r1, 0)≥ 0
I About the condition (P):
(E)
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity(quasipositivity of f ): ∀i = 1, ...,m∀r = (r1, ..., rm) ∈ [0,∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
I ’Formal’ proof: we extend f to IRm by fi (r) := fi (r +) so thatfi (r1, ..., ri−1, 0, ri+1, ..., rm) = fi (r
+1 , ..., r
+i−1, 0, r
+i+1, ..., r
+m ) ≥ 0, ∀r ∈ IRm
I Multiply the i-th equation by u−i := sup−ui , 0 = (−ui )+ and integrate:∫Ω
u−i ∂tui − diu−i ∆ui =
∫Ω
u−i fi (u).
I Using ∂t(u−i )2 = −2u−i ∂tui ,∇(u−i )2 = −2u−i ∇ui ,
−∫
Ω∂t(u−i )2+di |∇(u−i )2| = 2
∫Ω
u−i fi (u) = 2
∫Ω
u−i fi (u+)≥ 0 ,
I ⇒ ∂t∫
Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.
I About the condition (P):
(E)
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity(quasipositivity of f ): ∀i = 1, ...,m∀r = (r1, ..., rm) ∈ [0,∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
I ’Formal’ proof: we extend f to IRm by fi (r) := fi (r +) so thatfi (r1, ..., ri−1, 0, ri+1, ..., rm) = fi (r
+1 , ..., r
+i−1, 0, r
+i+1, ..., r
+m ) ≥ 0, ∀r ∈ IRm
I Multiply the i-th equation by u−i := sup−ui , 0 = (−ui )+ and integrate:∫Ω
u−i ∂tui − diu−i ∆ui =
∫Ω
u−i fi (u).
I Using ∂t(u−i )2 = −2u−i ∂tui ,∇(u−i )2 = −2u−i ∇ui ,
−∫
Ω∂t(u−i )2+di |∇(u−i )2| = 2
∫Ω
u−i fi (u) = 2
∫Ω
u−i fi (u+)≥ 0 ,
I ⇒ ∂t∫
Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.
I About the condition (P):
(E)
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity(quasipositivity of f ): ∀i = 1, ...,m∀r = (r1, ..., rm) ∈ [0,∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
I ’Formal’ proof: we extend f to IRm by fi (r) := fi (r +) so thatfi (r1, ..., ri−1, 0, ri+1, ..., rm) = fi (r
+1 , ..., r
+i−1, 0, r
+i+1, ..., r
+m ) ≥ 0, ∀r ∈ IRm
I Multiply the i-th equation by u−i := sup−ui , 0 = (−ui )+ and integrate:∫Ω
u−i ∂tui − diu−i ∆ui =
∫Ω
u−i fi (u).
I Using ∂t(u−i )2 = −2u−i ∂tui ,∇(u−i )2 = −2u−i ∇ui ,
−∫
Ω∂t(u−i )2+di |∇(u−i )2| = 2
∫Ω
u−i fi (u) = 2
∫Ω
u−i fi (u+)≥ 0 ,
I ⇒ ∂t∫
Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.
I About the condition (P):
(E)
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity(quasipositivity of f ): ∀i = 1, ...,m∀r = (r1, ..., rm) ∈ [0,∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
I ’Formal’ proof: we extend f to IRm by fi (r) := fi (r +) so thatfi (r1, ..., ri−1, 0, ri+1, ..., rm) = fi (r
+1 , ..., r
+i−1, 0, r
+i+1, ..., r
+m ) ≥ 0, ∀r ∈ IRm
I Multiply the i-th equation by u−i := sup−ui , 0 = (−ui )+ and integrate:∫Ω
u−i ∂tui − diu−i ∆ui =
∫Ω
u−i fi (u).
I Using ∂t(u−i )2 = −2u−i ∂tui ,∇(u−i )2 = −2u−i ∇ui ,
−∫
Ω∂t(u−i )2+di |∇(u−i )2| = 2
∫Ω
u−i fi (u) = 2
∫Ω
u−i fi (u+)≥ 0 ,
I ⇒ ∂t∫
Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.
I About the condition (M) :∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity ∀i = 1, ...,m∀r ∈ [0,+∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
I (M):∑
1≤i≤m fi (r1, ..., rm) ≤ 0 ⇒ ’Control of the TotalMass’:∫
Ω
∑1≤i≤m
ui (t, x)dx ≤∫
Ω
∑1≤i≤m
u0i (x)dx , ∀t ≥ 0.
Add up the m equations, integrate on Ω, use∫
Ω∆ui =
∫∂Ω∂νui = 0:∫
Ω
∂t [∑
ui (t)]dx =
∫Ω
∑i
fi (u)dx ≤ 0.
I ⇒ L1(Ω)- a priori estimates, uniform in time (t ∈ [0,T ∗) ).
I About the condition (M) :∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I (P) Preservation of Positivity ∀i = 1, ...,m∀r ∈ [0,+∞[m, fi (r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.
I (M):∑
1≤i≤m fi (r1, ..., rm) ≤ 0 ⇒ ’Control of the TotalMass’:∫
Ω
∑1≤i≤m
ui (t, x)dx ≤∫
Ω
∑1≤i≤m
u0i (x)dx , ∀t ≥ 0.
Add up the m equations, integrate on Ω, use∫
Ω∆ui =
∫∂Ω∂νui = 0:∫
Ω
∂t [∑
ui (t)]dx =
∫Ω
∑i
fi (u)dx ≤ 0.
I ⇒ L1(Ω)- a priori estimates, uniform in time (t ∈ [0,T ∗) ).
Same with (P)+(M’).
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I The L1-estimates, and most of the coming analysis carry overto (P) + ”natural” generalization of (M)
I (M’)∑
1≤i≤m ai fi (r) ≤ K0 + K1∑
1≤i≤m ri , ∀ r ∈ [0,+∞)m,for some ai ∈ (0,+∞) and K0,K1 ∈ IR.
I ⇒ ‖ui (t)‖L1(Ω) ≤ C(T , ‖u0‖L1(Ω)m
), ∀t ∈ [0,T ],∀T < +∞
I and we can also make f depend regularly on time and space( f = f (t, x , r) ).
Same with (P)+(M’).
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I The L1-estimates, and most of the coming analysis carry overto (P) + ”natural” generalization of (M)
I (M’)∑
1≤i≤m ai fi (r) ≤ K0 + K1∑
1≤i≤m ri , ∀ r ∈ [0,+∞)m,for some ai ∈ (0,+∞) and K0,K1 ∈ IR .
I ⇒ ‖ui (t)‖L1(Ω) ≤ C(T , ‖u0‖L1(Ω)m
), ∀t ∈ [0,T ],∀T < +∞
I and we can also make f depend regularly on time and space( f = f (t, x , r) ).
Same with (P)+(M’).
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I The L1-estimates, and most of the coming analysis carry overto (P) + ”natural” generalization of (M)
I (M’)∑
1≤i≤m ai fi (r) ≤ K0 + K1∑
1≤i≤m ri , ∀ r ∈ [0,+∞)m,for some ai ∈ (0,+∞) and K0,K1 ∈ IR .
I ⇒ ‖ui (t)‖L1(Ω) ≤ C(T , ‖u0‖L1(Ω)m
), ∀t ∈ [0,T ],∀T < +∞
I and we can also make f depend regularly on time and space( f = f (t, x , r) ).
Same with (P)+(M’).
∀i = 1, ...,m∂tui − di∆ui = fi (u1, u2, ..., um) in QT
∂νui = 0 on ΣT
ui (0, ·) = u0i (·) ≥ 0.
I The L1-estimates, and most of the coming analysis carry overto (P) + ”natural” generalization of (M)
I (M’)∑
1≤i≤m ai fi (r) ≤ K0 + K1∑
1≤i≤m ri , ∀ r ∈ [0,+∞)m,for some ai ∈ (0,+∞) and K0,K1 ∈ IR .
I ⇒ ‖ui (t)‖L1(Ω) ≤ C(T , ‖u0‖L1(Ω)m
), ∀t ∈ [0,T ],∀T < +∞
I and we can also make f depend regularly on time and space( f = f (t, x , r) ).
EXERCISE: global existence of solutions for the O.D.E. ?
I We assume (P)+(M). Does global existence hold in
(ODE )
∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0
i ∈ [0,+∞) ???
I By Cauchy-Lipschitz Theorem, we have existence of a localsolution on some maximal interval [0,T ∗),T ∗ ≤ +∞ and[supt∈[0,T∗) ui (t) < +∞,∀1 ≤ i ≤ m
]⇒ [T ∗ +∞].
I (P) ⇒ This solution is nonnegative.
I (M) ⇒ ∂t (∑
i ui (t)) =∑
i fi (u) ≤ 0.
I ⇒∑
i ui (t) ≤∑
i u0i , ∀t ∈ [0,T ∗)
I Together with nonnegativity ⇒ No blow up at T ∗ !⇒ T ∗ = +∞.
EXERCISE: global existence of solutions for the O.D.E. ?
I We assume (P)+(M). Does global existence hold in
(ODE )
∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0
i ∈ [0,+∞) ???
I By Cauchy-Lipschitz Theorem, we have existence of a localsolution on some maximal interval [0,T ∗),T ∗ ≤ +∞ and[supt∈[0,T∗) ui (t) < +∞,∀1 ≤ i ≤ m
]⇒ [T ∗ +∞].
I (P) ⇒ This solution is nonnegative.
I (M) ⇒ ∂t (∑
i ui (t)) =∑
i fi (u) ≤ 0.
I ⇒∑
i ui (t) ≤∑
i u0i , ∀t ∈ [0,T ∗)
I Together with nonnegativity ⇒ No blow up at T ∗ !⇒ T ∗ = +∞.
EXERCISE: global existence of solutions for the O.D.E. ?
I We assume (P)+(M). Does global existence hold in
(ODE )
∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0
i ∈ [0,+∞) ???
I By Cauchy-Lipschitz Theorem, we have existence of a localsolution on some maximal interval [0,T ∗),T ∗ ≤ +∞ and[supt∈[0,T∗) ui (t) < +∞,∀1 ≤ i ≤ m
]⇒ [T ∗ +∞].
I (P) ⇒ This solution is nonnegative.
I (M) ⇒ ∂t (∑
i ui (t)) =∑
i fi (u) ≤ 0.
I ⇒∑
i ui (t) ≤∑
i u0i , ∀t ∈ [0,T ∗)
I Together with nonnegativity ⇒ No blow up at T ∗ !⇒ T ∗ = +∞.
EXERCISE: global existence of solutions for the O.D.E. ?
I We assume (P)+(M). Does global existence hold in
(ODE )
∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0
i ∈ [0,+∞) ???
I By Cauchy-Lipschitz Theorem, we have existence of a localsolution on some maximal interval [0,T ∗),T ∗ ≤ +∞ and[supt∈[0,T∗) ui (t) < +∞,∀1 ≤ i ≤ m
]⇒ [T ∗ +∞].
I (P) ⇒ This solution is nonnegative.
I (M) ⇒ ∂t (∑
i ui (t)) =∑
i fi (u) ≤ 0.
I ⇒∑
i ui (t) ≤∑
i u0i , ∀t ∈ [0,T ∗)
I Together with nonnegativity ⇒ No blow up at T ∗ !⇒ T ∗ = +∞.
EXERCISE: global existence of solutions for the O.D.E. ?
I We assume (P)+(M). Does global existence hold in
(ODE )
∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0
i ∈ [0,+∞) ???
I By Cauchy-Lipschitz Theorem, we have existence of a localsolution on some maximal interval [0,T ∗),T ∗ ≤ +∞ and[supt∈[0,T∗) ui (t) < +∞,∀1 ≤ i ≤ m
]⇒ [T ∗ +∞].
I (P) ⇒ This solution is nonnegative.
I (M) ⇒ ∂t (∑
i ui (t)) =∑
i fi (u) ≤ 0.
I ⇒∑
i ui (t) ≤∑
i u0i , ∀t ∈ [0,T ∗)
I Together with nonnegativity ⇒ No blow up at T ∗ !⇒ T ∗ = +∞.
EXERCISE: global existence of solutions for the O.D.E. ?
I We assume (P)+(M). Does global existence hold in
(ODE )
∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0
i ∈ [0,+∞) ???
I By Cauchy-Lipschitz Theorem, we have existence of a localsolution on some maximal interval [0,T ∗),T ∗ ≤ +∞ and[supt∈[0,T∗) ui (t) < +∞,∀1 ≤ i ≤ m
]⇒ [T ∗ +∞].
I (P) ⇒ This solution is nonnegative.
I (M) ⇒ ∂t (∑
i ui (t)) =∑
i fi (u) ≤ 0.
I ⇒∑
i ui (t) ≤∑
i u0i , ∀t ∈ [0,T ∗)
I Together with nonnegativity ⇒ No blow up at T ∗ !⇒ T ∗ = +∞.
Local existence holds for the full RD-system
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I There is a ”Cauchy-Lipschitz Theorem” of local existencefor the PDE system, namely, for u0 ∈ L∞(Ω)+m (by afixed-point argument in L∞(QT )).
I There exists a maximal time T ∗ ∈ (0,+∞] and a(unique) classical solution u to (ST∗) with[
supt∈[0,T∗)
‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,
]⇒ [T ∗ = +∞]
I (P) ⇒ The solution u is nonnegative.
I (M) ⇒ supt∈[0,T∗) ‖ui(t)‖L1(Ω) < +∞,∀ i .But not sufficient to provide global existence.
Local existence holds for the full RD-system
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I There is a ”Cauchy-Lipschitz Theorem” of local existencefor the PDE system, namely, for u0 ∈ L∞(Ω)+m (by afixed-point argument in L∞(QT )).
I There exists a maximal time T ∗ ∈ (0,+∞] and a(unique) classical solution u to (ST∗) with[
supt∈[0,T∗)
‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,
]⇒ [T ∗ = +∞]
I (P) ⇒ The solution u is nonnegative.
I (M) ⇒ supt∈[0,T∗) ‖ui(t)‖L1(Ω) < +∞,∀ i .But not sufficient to provide global existence.
Local existence holds for the full RD-system
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I There is a ”Cauchy-Lipschitz Theorem” of local existencefor the PDE system, namely, for u0 ∈ L∞(Ω)+m (by afixed-point argument in L∞(QT )).
I There exists a maximal time T ∗ ∈ (0,+∞] and a(unique) classical solution u to (ST∗) with[
supt∈[0,T∗)
‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,
]⇒ [T ∗ = +∞]
I (P) ⇒ The solution u is nonnegative.
I (M) ⇒ supt∈[0,T∗) ‖ui(t)‖L1(Ω) < +∞,∀ i .But not sufficient to provide global existence.
Local existence holds for the full RD-system
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I There is a ”Cauchy-Lipschitz Theorem” of local existencefor the PDE system, namely, for u0 ∈ L∞(Ω)+m (by afixed-point argument in L∞(QT )).
I There exists a maximal time T ∗ ∈ (0,+∞] and a(unique) classical solution u to (ST∗) with[
supt∈[0,T∗)
‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,
]⇒ [T ∗ = +∞]
I (P) ⇒ The solution u is nonnegative.
I (M) ⇒ supt∈[0,T∗) ‖ui(t)‖L1(Ω) < +∞,∀ i .But not sufficient to provide global existence.
EXERCISE: Global existence when di = d ,∀i
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I We assume di = d ∈ (0,+∞), ∀i = 1, ...,m. Prove thatglobal existence holds under (P)+(M).
I By summing the m equations, we have
∂t
(∑i
ui
)− d∆
(∑i
ui
)=∑i
fi(u)≤ 0.
I By maximum principle for the heat equation, we deduce
‖∑i
ui(t)‖L∞(Ω) ≤ ‖∑i
u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).
I By nonnegativity, supt∈[0,T∗) ‖ui(t)‖L∞(Ω) < +∞,∀i
⇒ T ∗ = +∞.
EXERCISE: Global existence when di = d ,∀i
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I We assume di = d ∈ (0,+∞), ∀i = 1, ...,m. Prove thatglobal existence holds under (P)+(M).
I By summing the m equations, we have
∂t
(∑i
ui
)− d∆
(∑i
ui
)=∑i
fi(u)≤ 0.
I By maximum principle for the heat equation, we deduce
‖∑i
ui(t)‖L∞(Ω) ≤ ‖∑i
u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).
I By nonnegativity, supt∈[0,T∗) ‖ui(t)‖L∞(Ω) < +∞,∀i
⇒ T ∗ = +∞.
EXERCISE: Global existence when di = d ,∀i
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I We assume di = d ∈ (0,+∞), ∀i = 1, ...,m. Prove thatglobal existence holds under (P)+(M).
I By summing the m equations, we have
∂t
(∑i
ui
)− d∆
(∑i
ui
)=∑i
fi(u)≤ 0.
I By maximum principle for the heat equation, we deduce
‖∑i
ui(t)‖L∞(Ω) ≤ ‖∑i
u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).
I By nonnegativity, supt∈[0,T∗) ‖ui(t)‖L∞(Ω) < +∞,∀i
⇒ T ∗ = +∞.
EXERCISE: Global existence when di = d ,∀i
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I We assume di = d ∈ (0,+∞), ∀i = 1, ...,m. Prove thatglobal existence holds under (P)+(M).
I By summing the m equations, we have
∂t
(∑i
ui
)− d∆
(∑i
ui
)=∑i
fi(u)≤ 0.
I By maximum principle for the heat equation, we deduce
‖∑i
ui(t)‖L∞(Ω) ≤ ‖∑i
u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).
I By nonnegativity, supt∈[0,T∗) ‖ui(t)‖L∞(Ω) < +∞,∀i
⇒ T ∗ = +∞.
Examples of finite time blow up with (P)+(M)
I When the di are different from each other, it may happenthat L∞-blow up occurs in finite time for 2× 2 systemswith the two properties (P)+(M).
I Blow up can happen even in space dimension N = 1. Forthis we need to choose the nonlinearity f = f (r1, r2)growing very fast as r1 + r2 → +∞.
I As soon as the growth of f is more than quadratic asr1 + r2 → +∞, then blow up may occur in high spacedimension N .
I Let us state the result [M.P.-D. Schmitt].
Examples of finite time blow up with (P)+(M)
I When the di are different from each other, it may happenthat L∞-blow up occurs in finite time for 2× 2 systemswith the two properties (P)+(M).
I Blow up can happen even in space dimension N = 1. Forthis we need to choose the nonlinearity f = f (r1, r2)growing very fast as r1 + r2 → +∞.
I As soon as the growth of f is more than quadratic asr1 + r2 → +∞, then blow up may occur in high spacedimension N .
I Let us state the result [M.P.-D. Schmitt].
Examples of finite time blow up with (P)+(M)
I When the di are different from each other, it may happenthat L∞-blow up occurs in finite time for 2× 2 systemswith the two properties (P)+(M).
I Blow up can happen even in space dimension N = 1. Forthis we need to choose the nonlinearity f = f (r1, r2)growing very fast as r1 + r2 → +∞.
I As soon as the growth of f is more than quadratic asr1 + r2 → +∞, then blow up may occur in high spacedimension N .
I Let us state the result [M.P.-D. Schmitt].
Examples of finite time blow up with (P)+(M)
I When the di are different from each other, it may happenthat L∞-blow up occurs in finite time for 2× 2 systemswith the two properties (P)+(M).
I Blow up can happen even in space dimension N = 1. Forthis we need to choose the nonlinearity f = f (r1, r2)growing very fast as r1 + r2 → +∞.
I As soon as the growth of f is more than quadratic asr1 + r2 → +∞, then blow up may occur in high spacedimension N .
I Let us state the result [M.P.-D. Schmitt].
Examples of finite time blow up with (P)+(M)I Let η > 0 and T > 0. Then there exists
f1, f2 ∈ C∞([0,+∞)2) satisfying (P)+(M) and
(|f1|+ |f2|)(r1, r2) ≤ C [1 + (r1 + r2)2+η], ∀(r1, r2) ∈ [0,+∞)2
and di , αi , u0i such that the solution of the following system
with Ω = BN , the unit ball in IRN and N large enoughfor i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
satisfies : limt→T−
‖ui (t)‖L∞(Ω) = +∞, i = 1, 2.
I To be compared with the recent positive result: there existε = ε(N) > 0 such that global existence of classical solutionsholds if the growth of f is less than 2 + ε, together with(P)+(M) (to be described later).
Examples of finite time blow up with (P)+(M)I Let η > 0 and T > 0. Then there exists
f1, f2 ∈ C∞([0,+∞)2) satisfying (P)+(M) and
(|f1|+ |f2|)(r1, r2) ≤ C [1 + (r1 + r2)2+η], ∀(r1, r2) ∈ [0,+∞)2
and di , αi , u0i such that the solution of the following system
with Ω = BN , the unit ball in IRN and N large enoughfor i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
satisfies : limt→T−
‖ui (t)‖L∞(Ω) = +∞, i = 1, 2.
I To be compared with the recent positive result: there existε = ε(N) > 0 such that global existence of classical solutionsholds if the growth of f is less than 2 + ε, together with(P)+(M) (to be described later).
Examples of finite time blow up with (P)+(M)I Comments on these examples:
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I These solutions blow up at time t = T even in Lq(Ω) forq > N(1 + η)/2.
I They can be extended as global weak solutions up to t = +∞.
I We can adapt the examples to obtain homogeneous Neumannboundary conditions, but with f = f (t, x , r).
I The same examples provides also finite-time blow up for
∂tui − di∆ui = ci (t, x)uα1 uβ2 , i = 1, 2
with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.
Examples of finite time blow up with (P)+(M)I Comments on these examples:
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I These solutions blow up at time t = T even in Lq(Ω) forq > N(1 + η)/2.
I They can be extended as global weak solutions up to t = +∞.
I We can adapt the examples to obtain homogeneous Neumannboundary conditions, but with f = f (t, x , r).
I The same examples provides also finite-time blow up for
∂tui − di∆ui = ci (t, x)uα1 uβ2 , i = 1, 2
with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.
Examples of finite time blow up with (P)+(M)I Comments on these examples:
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I These solutions blow up at time t = T even in Lq(Ω) forq > N(1 + η)/2.
I They can be extended as global weak solutions up to t = +∞.
I We can adapt the examples to obtain homogeneous Neumannboundary conditions, but with f = f (t, x , r).
I The same examples provides also finite-time blow up for
∂tui − di∆ui = ci (t, x)uα1 uβ2 , i = 1, 2
with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.
Examples of finite time blow up with (P)+(M)I Comments on these examples:
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I These solutions blow up at time t = T even in Lq(Ω) forq > N(1 + η)/2.
I They can be extended as global weak solutions up to t = +∞.
I We can adapt the examples to obtain homogeneous Neumannboundary conditions, but with f = f (t, x , r).
I The same examples provides also finite-time blow up for
∂tui − di∆ui = ci (t, x)uα1 uβ2 , i = 1, 2
with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.
Examples of finite time blow up with (P)+(M)I Comments on these examples:
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I These solutions blow up at time t = T even in Lq(Ω) forq > N(1 + η)/2.
I They can be extended as global weak solutions up to t = +∞.
I We can adapt the examples to obtain homogeneous Neumannboundary conditions, but with f = f (t, x , r).
I The same examples provides also finite-time blow up for
∂tui − di∆ui = ci (t, x)uα1 uβ2 , i = 1, 2
with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I Note that ui is C∞ on [0,T ) and ui (T , x) = bi |x |2(1−γ).Thus limt→T− ‖ui (t)‖L∞(BN) = +∞,
I We choose the 8 parameters ai , bi , di , γ,N so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I CHOICE: d1 = N1/2, d2 = N−3, γ = 2− θ/N, θ ∈ (4/3,+∞),a1 = N−2, b1 = b2 = N−1/2, a2 = 2N − N−2 and N large.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I Note that ui is C∞ on [0,T ) and ui (T , x) = bi |x |2(1−γ).Thus limt→T− ‖ui (t)‖L∞(BN) = +∞,
I We choose the 8 parameters ai , bi , di , γ,N so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I CHOICE: d1 = N1/2, d2 = N−3, γ = 2− θ/N, θ ∈ (4/3,+∞),a1 = N−2, b1 = b2 = N−1/2, a2 = 2N − N−2 and N large.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I Note that ui is C∞ on [0,T ) and ui (T , x) = bi |x |2(1−γ).Thus limt→T− ‖ui (t)‖L∞(BN) = +∞,
I We choose the 8 parameters ai , bi , di , γ,N so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I CHOICE: d1 = N1/2, d2 = N−3, γ = 2− θ/N, θ ∈ (4/3,+∞),a1 = N−2, b1 = b2 = N−1/2, a2 = 2N − N−2 and N large.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I Note that ui is C∞ on [0,T ) and ui (T , x) = bi |x |2(1−γ).Thus limt→T− ‖ui (t)‖L∞(BN) = +∞,
I We choose the 8 parameters ai , bi , di , γ,N so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I CHOICE: d1 = N1/2, d2 = N−3, γ = 2− θ/N, θ ∈ (4/3,+∞),a1 = N−2, b1 = b2 = N−1/2, a2 = 2N − N−2 and N large.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The 8 parameters ai , bi , di , γ,N are chosen so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I We prove that ∂tui − di∆ui is of the form fi (u1, u2) with therequired properties...and with homogeneity γ′.
I Here γ′ = 2 + θN−θ , θ ∈ (4/3,+∞) close to 2 for large N.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The 8 parameters ai , bi , di , γ,N are chosen so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I We prove that ∂tui − di∆ui is of the form fi (u1, u2) with therequired properties...and with homogeneity γ′.
I Here γ′ = 2 + θN−θ , θ ∈ (4/3,+∞) close to 2 for large N.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The 8 parameters ai , bi , di , γ,N are chosen so that
∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.
I We prove that ∂tui − di∆ui is of the form fi (u1, u2) with therequired properties...and with homogeneity γ′.
I Here γ′ = 2 + θN−θ , θ ∈ (4/3,+∞) close to 2 for large N.
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The same kind of computation provides blowing up examplesin any dimension, including small dimensions as well.
I In dimension N = 1, we can have blow up with fi of degree 6.
I In dimension N = 2 with degree 7/2.
I In dimension N = 3 with degree 3 and even 3− ε !
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The same kind of computation provides blowing up examplesin any dimension, including small dimensions as well.
I In dimension N = 1, we can have blow up with fi of degree 6.
I In dimension N = 2 with degree 7/2.
I In dimension N = 3 with degree 3 and even 3− ε !
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The same kind of computation provides blowing up examplesin any dimension, including small dimensions as well.
I In dimension N = 1, we can have blow up with fi of degree 6.
I In dimension N = 2 with degree 7/2.
I In dimension N = 3 with degree 3 and even 3− ε !
Finite time blow up with (P)+(M): ideas of the proof
for i = 1, 2,∂tui − di∆ui = fi (u1, u2) in QT ,ui = αi on ΣT , αi ∈ C∞([0,T ])+,
ui (0, ·) = u0i ∈ C∞(Ω)+,
I Main idea: work with a priori solutions u1, u2 of the form
ui (t, x) =ai (T − t) + bi |x |2
(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.
I The same kind of computation provides blowing up examplesin any dimension, including small dimensions as well.
I In dimension N = 1, we can have blow up with fi of degree 6.
I In dimension N = 2 with degree 7/2.
I In dimension N = 3 with degree 3 and even 3− ε !
Global existence through the Lp-duality technique :”triangular” systems, and closed di
I This Lp-technique is useful in the study of global existence,but also for the asymptotic behavior of the solutions.
I Let us consider the (P)+(M) system where α, β ∈ [1,+∞)∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I Since u1 ≥ 0 and ∂tu1 − d1∆u1 ≤ 0, by maximum principle
‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).
I Since ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1), we formally have
u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).
I Main fact: A : Lp(QT )→ Lp(QT ) continuous ∀p ∈ (1,+∞)
‖u2‖Lp(QT ) ≤ C [1+‖u1‖Lp(QT )], C = C (‖u0i ‖Lp(Ω), p,T ,N,Ω, di ).
Global existence through the Lp-duality technique :”triangular” systems, and closed di
I This Lp-technique is useful in the study of global existence,but also for the asymptotic behavior of the solutions.
I Let us consider the (P)+(M) system where α, β ∈ [1,+∞)∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I Since u1 ≥ 0 and ∂tu1 − d1∆u1 ≤ 0, by maximum principle
‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).
I Since ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1), we formally have
u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).
I Main fact: A : Lp(QT )→ Lp(QT ) continuous ∀p ∈ (1,+∞)
‖u2‖Lp(QT ) ≤ C [1+‖u1‖Lp(QT )], C = C (‖u0i ‖Lp(Ω), p,T ,N,Ω, di ).
Global existence through the Lp-duality technique :”triangular” systems, and closed di
I This Lp-technique is useful in the study of global existence,but also for the asymptotic behavior of the solutions.
I Let us consider the (P)+(M) system where α, β ∈ [1,+∞)∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I Since u1 ≥ 0 and ∂tu1 − d1∆u1 ≤ 0, by maximum principle
‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).
I Since ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1), we formally have
u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).
I Main fact: A : Lp(QT )→ Lp(QT ) continuous ∀p ∈ (1,+∞)
‖u2‖Lp(QT ) ≤ C [1+‖u1‖Lp(QT )], C = C (‖u0i ‖Lp(Ω), p,T ,N,Ω, di ).
Global existence through the Lp-duality technique :”triangular” systems, and closed di
I This Lp-technique is useful in the study of global existence,but also for the asymptotic behavior of the solutions.
I Let us consider the (P)+(M) system where α, β ∈ [1,+∞)∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I Since u1 ≥ 0 and ∂tu1 − d1∆u1 ≤ 0, by maximum principle
‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).
I Since ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1), we formally have
u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).
I Main fact: A : Lp(QT )→ Lp(QT ) continuous ∀p ∈ (1,+∞)
‖u2‖Lp(QT ) ≤ C [1+‖u1‖Lp(QT )], C = C (‖u0i ‖Lp(Ω), p,T ,N,Ω, di ).
Global existence through the Lp-duality technique :”triangular” systems, and closed di
I This Lp-technique is useful in the study of global existence,but also for the asymptotic behavior of the solutions.
I Let us consider the (P)+(M) system where α, β ∈ [1,+∞)∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I Since u1 ≥ 0 and ∂tu1 − d1∆u1 ≤ 0, by maximum principle
‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).
I Since ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1), we formally have
u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).
I Main fact: A : Lp(QT )→ Lp(QT ) continuous ∀p ∈ (1,+∞)
‖u2‖Lp(QT ) ≤ C [1+‖u1‖Lp(QT )], C = C (‖u0i ‖Lp(Ω), p,T ,N,Ω, di ).
Global existence through the Lp-duality technique :”triangular” systems, and closed di
End of the proof of existence for all T > 0 for the ”triangular”system:
∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,
∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I We know that ‖u1(t)‖L∞(QT ) is uniformly bounded.
I By the ’Main fact’, for all p ∈ (1,+∞), and all T > 0,‖u2‖Lp(QT ) ≤ C [1 + ‖u1‖Lp(QT )] ≤ C1[1 + ‖u1‖L∞(QT )] ≤ C2.
I ‖uα1 uβ2 ‖Lq(QT ) ≤ C‖u1‖αL∞(Ω)‖u2‖βLqβ(QT )≤ C3.
I By the heat equation in u2, choosing q large implies a boundon the L∞(QT ) norm of u2, this for all T > 0. Whence theglobal existence for the above system.
Global existence through the Lp-duality technique :”triangular” systems, and closed di
End of the proof of existence for all T > 0 for the ”triangular”system:
∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,
∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I We know that ‖u1(t)‖L∞(QT ) is uniformly bounded.
I By the ’Main fact’, for all p ∈ (1,+∞), and all T > 0,‖u2‖Lp(QT ) ≤ C [1 + ‖u1‖Lp(QT )] ≤ C1[1 + ‖u1‖L∞(QT )] ≤ C2.
I ‖uα1 uβ2 ‖Lq(QT ) ≤ C‖u1‖αL∞(Ω)‖u2‖βLqβ(QT )≤ C3.
I By the heat equation in u2, choosing q large implies a boundon the L∞(QT ) norm of u2, this for all T > 0. Whence theglobal existence for the above system.
Global existence through the Lp-duality technique :”triangular” systems, and closed di
End of the proof of existence for all T > 0 for the ”triangular”system:
∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,
∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I We know that ‖u1(t)‖L∞(QT ) is uniformly bounded.
I By the ’Main fact’, for all p ∈ (1,+∞), and all T > 0,‖u2‖Lp(QT ) ≤ C [1 + ‖u1‖Lp(QT )] ≤ C1[1 + ‖u1‖L∞(QT )] ≤ C2.
I ‖uα1 uβ2 ‖Lq(QT ) ≤ C‖u1‖αL∞(Ω)‖u2‖βLqβ(QT )≤ C3.
I By the heat equation in u2, choosing q large implies a boundon the L∞(QT ) norm of u2, this for all T > 0. Whence theglobal existence for the above system.
Global existence through the Lp-duality technique :”triangular” systems, and closed di
End of the proof of existence for all T > 0 for the ”triangular”system:
∂tu1 − d1∆u1 = −uα1 uβ2 in QT
∂tu2 − d2∆u2 = uα1 uβ2 in QT ,
∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0
i ≥ 0, i = 1, 2.
I We know that ‖u1(t)‖L∞(QT ) is uniformly bounded.
I By the ’Main fact’, for all p ∈ (1,+∞), and all T > 0,‖u2‖Lp(QT ) ≤ C [1 + ‖u1‖Lp(QT )] ≤ C1[1 + ‖u1‖L∞(QT )] ≤ C2.
I ‖uα1 uβ2 ‖Lq(QT ) ≤ C‖u1‖αL∞(Ω)‖u2‖βLqβ(QT )≤ C3.
I By the heat equation in u2, choosing q large implies a boundon the L∞(QT ) norm of u2, this for all T > 0. Whence theglobal existence for the above system.
EXERCICE: Prove the Lp(QT )-estimate by dualityI STATEMENT: Let u1, u2 satisfy u2 ≥ 0 and
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Prove that, for some C depending on u0i , but not on T
(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].
I For this, consider the solution of the dual problem− (∂tψ + d2∆ψ) = Θ ∈ C∞0 (QT ),Θ ≥ 0,∂νψ = 0 on ΣT , ψ(T ) = 0, ψ ≥ 0.
I The maximal Lq-regularity theory for the heat equation says
(∗∗) ‖∂tψ‖Lq(QT ) + ‖∆ψ‖Lq(QT ) ≤ C‖Θ‖Lq(QT ), ∀q ∈ (1,∞),
for some C = C (q, d2,Ω) independent of T .I The expected estimate (∗) is exactly the dual of (∗∗) with q = p′.
HINT: Multiply (Ineq) by ψ ≥ 0 and integrate by parts.
EXERCICE: Prove the Lp(QT )-estimate by dualityI STATEMENT: Let u1, u2 satisfy u2 ≥ 0 and
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Prove that, for some C depending on u0i , but not on T
(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].
I For this, consider the solution of the dual problem− (∂tψ + d2∆ψ) = Θ ∈ C∞0 (QT ),Θ ≥ 0,∂νψ = 0 on ΣT , ψ(T ) = 0, ψ ≥ 0.
I The maximal Lq-regularity theory for the heat equation says
(∗∗) ‖∂tψ‖Lq(QT ) + ‖∆ψ‖Lq(QT ) ≤ C‖Θ‖Lq(QT ), ∀q ∈ (1,∞),
for some C = C (q, d2,Ω) independent of T .I The expected estimate (∗) is exactly the dual of (∗∗) with q = p′.
HINT: Multiply (Ineq) by ψ ≥ 0 and integrate by parts.
EXERCICE: Prove the Lp(QT )-estimate by dualityI STATEMENT: Let u1, u2 satisfy u2 ≥ 0 and
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Prove that, for some C depending on u0i , but not on T
(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].
I For this, consider the solution of the dual problem− (∂tψ + d2∆ψ) = Θ ∈ C∞0 (QT ),Θ ≥ 0,∂νψ = 0 on ΣT , ψ(T ) = 0, ψ ≥ 0.
I The maximal Lq-regularity theory for the heat equation says
(∗∗) ‖∂tψ‖Lq(QT ) + ‖∆ψ‖Lq(QT ) ≤ C‖Θ‖Lq(QT ), ∀q ∈ (1,∞),
for some C = C (q, d2,Ω) independent of T .I The expected estimate (∗) is exactly the dual of (∗∗) with q = p′.
HINT: Multiply (Ineq) by ψ ≥ 0 and integrate by parts.
EXERCICE: Prove the Lp(QT )-estimate by dualityI STATEMENT: Let u1, u2 satisfy u2 ≥ 0 and
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Prove that, for some C depending on u0i , but not on T
(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].
I For this, consider the solution of the dual problem− (∂tψ + d2∆ψ) = Θ ∈ C∞0 (QT ),Θ ≥ 0,∂νψ = 0 on ΣT , ψ(T ) = 0, ψ ≥ 0.
I The maximal Lq-regularity theory for the heat equation says
(∗∗) ‖∂tψ‖Lq(QT ) + ‖∆ψ‖Lq(QT ) ≤ C‖Θ‖Lq(QT ), ∀q ∈ (1,∞),
for some C = C (q, d2,Ω) independent of T .
I The expected estimate (∗) is exactly the dual of (∗∗) with q = p′.
HINT: Multiply (Ineq) by ψ ≥ 0 and integrate by parts.
EXERCICE: Prove the Lp(QT )-estimate by dualityI STATEMENT: Let u1, u2 satisfy u2 ≥ 0 and
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Prove that, for some C depending on u0i , but not on T
(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].
I For this, consider the solution of the dual problem− (∂tψ + d2∆ψ) = Θ ∈ C∞0 (QT ),Θ ≥ 0,∂νψ = 0 on ΣT , ψ(T ) = 0, ψ ≥ 0.
I The maximal Lq-regularity theory for the heat equation says
(∗∗) ‖∂tψ‖Lq(QT ) + ‖∆ψ‖Lq(QT ) ≤ C‖Θ‖Lq(QT ), ∀q ∈ (1,∞),
for some C = C (q, d2,Ω) independent of T .I The expected estimate (∗) is exactly the dual of (∗∗) with q = p′.
HINT: Multiply (Ineq) by ψ ≥ 0 and integrate by parts.
The proof of the Lp-estimate
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give
−∫QT
u2(∂tψ + d2∆ψ) ≤∫
Ω
[u02 − a u0
1 ]ψ(0) +
∫QT
u1[−a∂tψ + b∆ψ],
I ⇒
∫QT
u2Θ ≤ ‖u02 − au0
1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)
+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).
I By maximal regularity for ψ, there exists C independent of T with(∗∗) ‖∂tψ‖Lp′ (QT ) + ‖∆ψ‖Lp′ (QT ) ≤ C‖Θ‖Lp′ (QT ),
I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T
0∂tψ‖Lp′ (Ω) ≤ T 1/p‖∂tψ‖Lp′ (QT ) ≤ CT 1/p‖Θ‖Lp′ (QT ).
I∫QT
u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C
]‖Θ‖Lp′ (QT ),
with C0 = ‖u02 − au0
1‖Lp(Ω) and for arbitrary Θ ≥ 0.
I Whence by duality
‖u2‖Lp(QT ) ≤ CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C .
The proof of the Lp-estimate
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give
−∫QT
u2(∂tψ + d2∆ψ) ≤∫
Ω
[u02 − a u0
1 ]ψ(0) +
∫QT
u1[−a∂tψ + b∆ψ],
I ⇒
∫QT
u2Θ ≤ ‖u02 − au0
1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)
+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).
I By maximal regularity for ψ, there exists C independent of T with(∗∗) ‖∂tψ‖Lp′ (QT ) + ‖∆ψ‖Lp′ (QT ) ≤ C‖Θ‖Lp′ (QT ),
I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T
0∂tψ‖Lp′ (Ω) ≤ T 1/p‖∂tψ‖Lp′ (QT ) ≤ CT 1/p‖Θ‖Lp′ (QT ).
I∫QT
u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C
]‖Θ‖Lp′ (QT ),
with C0 = ‖u02 − au0
1‖Lp(Ω) and for arbitrary Θ ≥ 0.
I Whence by duality
‖u2‖Lp(QT ) ≤ CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C .
The proof of the Lp-estimate
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give
−∫QT
u2(∂tψ + d2∆ψ) ≤∫
Ω
[u02 − a u0
1 ]ψ(0) +
∫QT
u1[−a∂tψ + b∆ψ],
I ⇒
∫QT
u2Θ ≤ ‖u02 − au0
1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)
+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).
I By maximal regularity for ψ, there exists C independent of T with(∗∗) ‖∂tψ‖Lp′ (QT ) + ‖∆ψ‖Lp′ (QT ) ≤ C‖Θ‖Lp′ (QT ),
I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T
0∂tψ‖Lp′ (Ω) ≤ T 1/p‖∂tψ‖Lp′ (QT ) ≤ CT 1/p‖Θ‖Lp′ (QT ).
I∫QT
u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C
]‖Θ‖Lp′ (QT ),
with C0 = ‖u02 − au0
1‖Lp(Ω) and for arbitrary Θ ≥ 0.
I Whence by duality
‖u2‖Lp(QT ) ≤ CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C .
The proof of the Lp-estimate
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give
−∫QT
u2(∂tψ + d2∆ψ) ≤∫
Ω
[u02 − a u0
1 ]ψ(0) +
∫QT
u1[−a∂tψ + b∆ψ],
I ⇒
∫QT
u2Θ ≤ ‖u02 − au0
1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)
+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).
I By maximal regularity for ψ, there exists C independent of T with(∗∗) ‖∂tψ‖Lp′ (QT ) + ‖∆ψ‖Lp′ (QT ) ≤ C‖Θ‖Lp′ (QT ),
I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T
0∂tψ‖Lp′ (Ω) ≤ T 1/p‖∂tψ‖Lp′ (QT ) ≤ CT 1/p‖Θ‖Lp′ (QT ).
I∫QT
u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C
]‖Θ‖Lp′ (QT ),
with C0 = ‖u02 − au0
1‖Lp(Ω) and for arbitrary Θ ≥ 0.
I Whence by duality
‖u2‖Lp(QT ) ≤ CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C .
The proof of the Lp-estimate
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give
−∫QT
u2(∂tψ + d2∆ψ) ≤∫
Ω
[u02 − a u0
1 ]ψ(0) +
∫QT
u1[−a∂tψ + b∆ψ],
I ⇒
∫QT
u2Θ ≤ ‖u02 − au0
1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)
+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).
I By maximal regularity for ψ, there exists C independent of T with(∗∗) ‖∂tψ‖Lp′ (QT ) + ‖∆ψ‖Lp′ (QT ) ≤ C‖Θ‖Lp′ (QT ),
I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T
0∂tψ‖Lp′ (Ω) ≤ T 1/p‖∂tψ‖Lp′ (QT ) ≤ CT 1/p‖Θ‖Lp′ (QT ).
I∫QT
u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C
]‖Θ‖Lp′ (QT ),
with C0 = ‖u02 − au0
1‖Lp(Ω) and for arbitrary Θ ≥ 0.
I Whence by duality
‖u2‖Lp(QT ) ≤ CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C .
The proof of the Lp-estimate
(Ineq)
∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ IR,∂νui = 0 on ΣT , i = 1, 2.
I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give
−∫QT
u2(∂tψ + d2∆ψ) ≤∫
Ω
[u02 − a u0
1 ]ψ(0) +
∫QT
u1[−a∂tψ + b∆ψ],
I ⇒
∫QT
u2Θ ≤ ‖u02 − au0
1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)
+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).
I By maximal regularity for ψ, there exists C independent of T with(∗∗) ‖∂tψ‖Lp′ (QT ) + ‖∆ψ‖Lp′ (QT ) ≤ C‖Θ‖Lp′ (QT ),
I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T
0∂tψ‖Lp′ (Ω) ≤ T 1/p‖∂tψ‖Lp′ (QT ) ≤ CT 1/p‖Θ‖Lp′ (QT ).
I∫QT
u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C
]‖Θ‖Lp′ (QT ),
with C0 = ‖u02 − au0
1‖Lp(Ω) and for arbitrary Θ ≥ 0.
I Whence by duality
‖u2‖Lp(QT ) ≤ CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C .
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I A first one is a generalization of the previous global existenceresult to m ×m so-called ”triangular systems”.
I Consider an m ×m system for which there exists a (lower)triangular invertible matrix Q with nonnegative entries and
Q f (r) ≤ [1 +∑i
ri ]b, ∀r ∈ [0,∞)m, for some b ∈ IRm,
and with an at most polynomial growth for f .I Then global existence of classical solutions holds for the
corresponding RD-system.
I Note that b = 0 and Q =
[1 01 1
]in the 2× 2 example.
I Case f1 = −u1h(u2), f2 = u1h(u2) open for fast growing h, except a few
cases [Benachour-Rebiai].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I A first one is a generalization of the previous global existenceresult to m ×m so-called ”triangular systems”.
I Consider an m ×m system for which there exists a (lower)triangular invertible matrix Q with nonnegative entries and
Q f (r) ≤ [1 +∑i
ri ]b, ∀r ∈ [0,∞)m, for some b ∈ IRm,
and with an at most polynomial growth for f .
I Then global existence of classical solutions holds for thecorresponding RD-system.
I Note that b = 0 and Q =
[1 01 1
]in the 2× 2 example.
I Case f1 = −u1h(u2), f2 = u1h(u2) open for fast growing h, except a few
cases [Benachour-Rebiai].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I A first one is a generalization of the previous global existenceresult to m ×m so-called ”triangular systems”.
I Consider an m ×m system for which there exists a (lower)triangular invertible matrix Q with nonnegative entries and
Q f (r) ≤ [1 +∑i
ri ]b, ∀r ∈ [0,∞)m, for some b ∈ IRm,
and with an at most polynomial growth for f .I Then global existence of classical solutions holds for the
corresponding RD-system.
I Note that b = 0 and Q =
[1 01 1
]in the 2× 2 example.
I Case f1 = −u1h(u2), f2 = u1h(u2) open for fast growing h, except a few
cases [Benachour-Rebiai].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I A first one is a generalization of the previous global existenceresult to m ×m so-called ”triangular systems”.
I Consider an m ×m system for which there exists a (lower)triangular invertible matrix Q with nonnegative entries and
Q f (r) ≤ [1 +∑i
ri ]b, ∀r ∈ [0,∞)m, for some b ∈ IRm,
and with an at most polynomial growth for f .I Then global existence of classical solutions holds for the
corresponding RD-system.
I Note that b = 0 and Q =
[1 01 1
]in the 2× 2 example.
I Case f1 = −u1h(u2), f2 = u1h(u2) open for fast growing h, except a few
cases [Benachour-Rebiai].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I A first one is a generalization of the previous global existenceresult to m ×m so-called ”triangular systems”.
I Consider an m ×m system for which there exists a (lower)triangular invertible matrix Q with nonnegative entries and
Q f (r) ≤ [1 +∑i
ri ]b, ∀r ∈ [0,∞)m, for some b ∈ IRm,
and with an at most polynomial growth for f .I Then global existence of classical solutions holds for the
corresponding RD-system.
I Note that b = 0 and Q =
[1 01 1
]in the 2× 2 example.
I Case f1 = −u1h(u2), f2 = u1h(u2) open for fast growing h, except a few
cases [Benachour-Rebiai].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I We simply write, for any d ∈ (0,+∞)
∂t(∑i
ui )−d∆(∑i
ui ) =∑i
(di−d)∆ui+∑i
fi ≤ ∆
(∑i
(di − d)ui
).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I We simply write, for any d ∈ (0,+∞)
∂t(∑i
ui )−d∆(∑i
ui ) =∑i
(di−d)∆ui+∑i
fi ≤ ∆
(∑i
(di − d)ui
).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I We simply write, for any d ∈ (0,+∞)
∂t(∑i
ui )−d∆(∑i
ui ) =∑i
(di−d)∆ui+∑i
fi ≤ ∆
(∑i
(di − d)ui
).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ C [T 1/p + (maxi |di − d |)‖∑
i ui‖Lp(QT )].
I Choose d so that C (maxi |di − d |) < 1 (di close enough)
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ CT 1/p[1− C (maxi |di − d |)]−1,∀p <∞.
I Whence the global existence for (ST ) for all T > 0.
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ C [T 1/p + (maxi |di − d |)‖∑
i ui‖Lp(QT )].
I Choose d so that C (maxi |di − d |) < 1 (di close enough)
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ CT 1/p[1− C (maxi |di − d |)]−1,∀p <∞.
I Whence the global existence for (ST ) for all T > 0.
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ C [T 1/p + (maxi |di − d |)‖∑
i ui‖Lp(QT )].
I Choose d so that C (maxi |di − d |) < 1 (di close enough)
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ CT 1/p[1− C (maxi |di − d |)]−1,∀p <∞.
I Whence the global existence for (ST ) for all T > 0.
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ C [T 1/p + (maxi |di − d |)‖∑
i ui‖Lp(QT )].
I Choose d so that C (maxi |di − d |) < 1 (di close enough)
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ CT 1/p[1− C (maxi |di − d |)]−1,∀p <∞.
I Whence the global existence for (ST ) for all T > 0.
Two main applications of the Lp approach
(ST )
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in QT ,∂νui = 0 on ΣT ,ui (0, ·) = u0
i (·) ≥ 0, u0i ∈ L∞(Ω)+.
I 2nd application: global existence of classical solutions if thedi , i = 1, ...,m are close enough to each other and thenonlinearity f is at most polynomial and satisfy (P)+(M).
I By the main Lp-estimate, for some C independent of T
‖∑i
ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i
(di − d)ui‖Lp(QT )].
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ C [T 1/p + (maxi |di − d |)‖∑
i ui‖Lp(QT )].
I Choose d so that C (maxi |di − d |) < 1 (di close enough)
I ⇒ ‖∑
i ui‖Lp(QT ) ≤ CT 1/p[1− C (maxi |di − d |)]−1,∀p <∞.
I Whence the global existence for (ST ) for all T > 0.
One more application of the Lp-approach
I The Lp-approach also applies to the following 3× 3 system∂tu1 − d1 ∆u1 = uα3
3 − uα11 uα2
2 ,∂tu2 − d2 ∆u2 = uα3
3 − uα11 uα2
2 ,∂tu3 − d3 ∆u3 = −uα3
3 + uα11 uα2
2 ,∂νui = 0 i = 1, 2, 3,ui (0, ·) = u0
i (·) ≥ 0, i = 1, 2, 3.
I It is proved that global existence of classical solutions holds ifα3 > α1 + α2 [El Haj Laamri].
I On the other hand, global existence is open in general ifα3 ≤ α1 + α2 (except in some cases for small values of theαi ).
One more application of the Lp-approach
I The Lp-approach also applies to the following 3× 3 system∂tu1 − d1 ∆u1 = uα3
3 − uα11 uα2
2 ,∂tu2 − d2 ∆u2 = uα3
3 − uα11 uα2
2 ,∂tu3 − d3 ∆u3 = −uα3
3 + uα11 uα2
2 ,∂νui = 0 i = 1, 2, 3,ui (0, ·) = u0
i (·) ≥ 0, i = 1, 2, 3.
I It is proved that global existence of classical solutions holds ifα3 > α1 + α2 [El Haj Laamri].
I On the other hand, global existence is open in general ifα3 ≤ α1 + α2 (except in some cases for small values of theαi ).
One more application of the Lp-approach
I The Lp-approach also applies to the following 3× 3 system∂tu1 − d1 ∆u1 = uα3
3 − uα11 uα2
2 ,∂tu2 − d2 ∆u2 = uα3
3 − uα11 uα2
2 ,∂tu3 − d3 ∆u3 = −uα3
3 + uα11 uα2
2 ,∂νui = 0 i = 1, 2, 3,ui (0, ·) = u0
i (·) ≥ 0, i = 1, 2, 3.
I It is proved that global existence of classical solutions holds ifα3 > α1 + α2 [El Haj Laamri].
I On the other hand, global existence is open in general ifα3 ≤ α1 + α2 (except in some cases for small values of theαi ).
Global existence of weak ”L1”-solutions
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I L1-THEOREM. Assume that (P)+(M) or +(M’) hold.Assume moreover that the following L1-a priori estimate holds:∫
QT
|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m.
Then there exists a global weak solution to (S∞), and this forall initial data u0
i ∈ L1(Ω)+, i = 1, ...,m.
I By a weak solution of (S∞), we mean that fi (u) ∈ L1(QT ) for allT > 0 and all i , and that the equations are satisfied (for instance)in the sense of the ”variation of constants” formula
ui (t) = Si (t)u0i +
∫ t
0
Si (t−s)fi (u(s))ds, i = 1, ...,m, ∀t ∈ [0,+∞),
where Si (t) is the semigroup generated in L1(Ω) by the operator
−di∆ with homogeneous Neumann boundary conditions.
Global existence of weak ”L1”-solutions
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I L1-THEOREM. Assume that (P)+(M) or +(M’) hold.Assume moreover that the following L1-a priori estimate holds:∫
QT
|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m.
Then there exists a global weak solution to (S∞), and this forall initial data u0
i ∈ L1(Ω)+, i = 1, ...,m.I By a weak solution of (S∞), we mean that fi (u) ∈ L1(QT ) for all
T > 0 and all i , and that the equations are satisfied (for instance)in the sense of the ”variation of constants” formula
ui (t) = Si (t)u0i +
∫ t
0
Si (t−s)fi (u(s))ds, i = 1, ...,m, ∀t ∈ [0,+∞),
where Si (t) is the semigroup generated in L1(Ω) by the operator
−di∆ with homogeneous Neumann boundary conditions.
Global existence of weak ”L1-solutions”
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I L1-THEOREM. What means the L1- a priori estimate:∫QT
|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m ?
I By an L1-a priori estimate, we mean that such an L1-estimateholds for the solutions of a good approximate system of (S∞).
I Let for instance f ni (r) := fi (r)
1+n−1∑
j |fj (r)| , i = 1, ...,m. Then
|f ni (r)| ≤ n. Thus the approximate system (Sn
∞), where fi isreplaced by f n
i in (S∞), has a global classical solution un.I Saying that the L1-a priori estimate holds means that
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, ∀i = 1, ...,m.
Global existence of weak ”L1-solutions”
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I L1-THEOREM. What means the L1- a priori estimate:∫QT
|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m ?
I By an L1-a priori estimate, we mean that such an L1-estimateholds for the solutions of a good approximate system of (S∞).
I Let for instance f ni (r) := fi (r)
1+n−1∑
j |fj (r)| , i = 1, ...,m. Then
|f ni (r)| ≤ n. Thus the approximate system (Sn
∞), where fi isreplaced by f n
i in (S∞), has a global classical solution un.
I Saying that the L1-a priori estimate holds means that
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, ∀i = 1, ...,m.
Global existence of weak ”L1-solutions”
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I L1-THEOREM. What means the L1- a priori estimate:∫QT
|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m ?
I By an L1-a priori estimate, we mean that such an L1-estimateholds for the solutions of a good approximate system of (S∞).
I Let for instance f ni (r) := fi (r)
1+n−1∑
j |fj (r)| , i = 1, ...,m. Then
|f ni (r)| ≤ n. Thus the approximate system (Sn
∞), where fi isreplaced by f n
i in (S∞), has a global classical solution un.I Saying that the L1-a priori estimate holds means that
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, ∀i = 1, ...,m.
Global existence of weak ”L1”-solutions: an exampleI Let the 2× 2 system where h : [0,+∞)→ [0,+∞), α ∈ [1,+∞)
∂tu1 − d1∆u1 = −uα1 h(u2) in Q∞,∂tu2 − d2∆u2 = uα1 h(u2) in Q∞,∂νui = 0 on Σ∞, ui (0) = u0
i ≥ 0, i = 1, 2.
I Global existence of classical solutions holds if the growth of his at most polynomial, and is OPEN for general h, for instancewith h(r) = er
q, q ≥ 2.
I But if we integrate the equation in u1 we have∫Ω
u1(T )− 0 +
∫QT
uα1 h(u2) =
∫Ω
u01 .
I This implies the a priori L1-bound on the nonlinear reaction:∫QT
uα1 h(u2) ≤∫
Ωu0
1 .
I By the L1-THEOREM, we can claim global existence of weaksolutions for this system.
I We will see more applications, but let’s give some ideas of the proof.
Global existence of weak ”L1”-solutions: an exampleI Let the 2× 2 system where h : [0,+∞)→ [0,+∞), α ∈ [1,+∞)
∂tu1 − d1∆u1 = −uα1 h(u2) in Q∞,∂tu2 − d2∆u2 = uα1 h(u2) in Q∞,∂νui = 0 on Σ∞, ui (0) = u0
i ≥ 0, i = 1, 2.
I Global existence of classical solutions holds if the growth of his at most polynomial, and is OPEN for general h, for instancewith h(r) = er
q, q ≥ 2.
I But if we integrate the equation in u1 we have∫Ω
u1(T )− 0 +
∫QT
uα1 h(u2) =
∫Ω
u01 .
I This implies the a priori L1-bound on the nonlinear reaction:∫QT
uα1 h(u2) ≤∫
Ωu0
1 .
I By the L1-THEOREM, we can claim global existence of weaksolutions for this system.
I We will see more applications, but let’s give some ideas of the proof.
Global existence of weak ”L1”-solutions: an exampleI Let the 2× 2 system where h : [0,+∞)→ [0,+∞), α ∈ [1,+∞)
∂tu1 − d1∆u1 = −uα1 h(u2) in Q∞,∂tu2 − d2∆u2 = uα1 h(u2) in Q∞,∂νui = 0 on Σ∞, ui (0) = u0
i ≥ 0, i = 1, 2.
I Global existence of classical solutions holds if the growth of his at most polynomial, and is OPEN for general h, for instancewith h(r) = er
q, q ≥ 2.
I But if we integrate the equation in u1 we have∫Ω
u1(T )− 0 +
∫QT
uα1 h(u2) =
∫Ω
u01 .
I This implies the a priori L1-bound on the nonlinear reaction:∫QT
uα1 h(u2) ≤∫
Ωu0
1 .
I By the L1-THEOREM, we can claim global existence of weaksolutions for this system.
I We will see more applications, but let’s give some ideas of the proof.
Global existence of weak ”L1”-solutions: an exampleI Let the 2× 2 system where h : [0,+∞)→ [0,+∞), α ∈ [1,+∞)
∂tu1 − d1∆u1 = −uα1 h(u2) in Q∞,∂tu2 − d2∆u2 = uα1 h(u2) in Q∞,∂νui = 0 on Σ∞, ui (0) = u0
i ≥ 0, i = 1, 2.
I Global existence of classical solutions holds if the growth of his at most polynomial, and is OPEN for general h, for instancewith h(r) = er
q, q ≥ 2.
I But if we integrate the equation in u1 we have∫Ω
u1(T )− 0 +
∫QT
uα1 h(u2) =
∫Ω
u01 .
I This implies the a priori L1-bound on the nonlinear reaction:∫QT
uα1 h(u2) ≤∫
Ωu0
1 .
I By the L1-THEOREM, we can claim global existence of weaksolutions for this system.
I We will see more applications, but let’s give some ideas of the proof.
Global existence of weak ”L1”-solutions: an exampleI Let the 2× 2 system where h : [0,+∞)→ [0,+∞), α ∈ [1,+∞)
∂tu1 − d1∆u1 = −uα1 h(u2) in Q∞,∂tu2 − d2∆u2 = uα1 h(u2) in Q∞,∂νui = 0 on Σ∞, ui (0) = u0
i ≥ 0, i = 1, 2.
I Global existence of classical solutions holds if the growth of his at most polynomial, and is OPEN for general h, for instancewith h(r) = er
q, q ≥ 2.
I But if we integrate the equation in u1 we have∫Ω
u1(T )− 0 +
∫QT
uα1 h(u2) =
∫Ω
u01 .
I This implies the a priori L1-bound on the nonlinear reaction:∫QT
uα1 h(u2) ≤∫
Ωu0
1 .
I By the L1-THEOREM, we can claim global existence of weaksolutions for this system.
I We will see more applications, but let’s give some ideas of the proof.
Global existence of weak ”L1”-solutions: an exampleI Let the 2× 2 system where h : [0,+∞)→ [0,+∞), α ∈ [1,+∞)
∂tu1 − d1∆u1 = −uα1 h(u2) in Q∞,∂tu2 − d2∆u2 = uα1 h(u2) in Q∞,∂νui = 0 on Σ∞, ui (0) = u0
i ≥ 0, i = 1, 2.
I Global existence of classical solutions holds if the growth of his at most polynomial, and is OPEN for general h, for instancewith h(r) = er
q, q ≥ 2.
I But if we integrate the equation in u1 we have∫Ω
u1(T )− 0 +
∫QT
uα1 h(u2) =
∫Ω
u01 .
I This implies the a priori L1-bound on the nonlinear reaction:∫QT
uα1 h(u2) ≤∫
Ωu0
1 .
I By the L1-THEOREM, we can claim global existence of weaksolutions for this system.
I We will see more applications, but let’s give some ideas of the proof.
Main ideas in the proof of the L1-THEOREM
I As already mentioned, we solve the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
where f ni (r) := fi (r)
1+n−1∑
j |fj (r)| , i = 1, ...,m. Since |f ni | is
bounded by n, and quasipositive, there exists such a classicalglobal nonnegative solution un, according to our firstlocal/global existence result.
I We assume that L1-estimates hold
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
Main ideas in the proof of the L1-THEOREM
I As already mentioned, we solve the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
where f ni (r) := fi (r)
1+n−1∑
j |fj (r)| , i = 1, ...,m. Since |f ni | is
bounded by n, and quasipositive, there exists such a classicalglobal nonnegative solution un, according to our firstlocal/global existence result.
I We assume that L1-estimates hold
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
Main ideas in the proof of the L1-THEOREM: first, anL1-compactness property
I The following mapping is compact :
L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
where w is the solution of∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I This is proved specifically in [Baras-Hassan-Veron] by semigroupstechniques. Or use that its dual is compact:
L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω),
where ψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Compactness of L∗ from the Lp-maximal regularity, p < +∞:
‖∂tψ‖Lp(QT )+‖∆ψ‖Lp(QT ) ≤ C‖Θ‖Lp(QT ) ≤ C (T )‖Θ‖L∞(QT ).
For p large, compact embedding into L∞(QT ) !
Main ideas in the proof of the L1-THEOREM: first, anL1-compactness property
I The following mapping is compact :
L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
where w is the solution of∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I This is proved specifically in [Baras-Hassan-Veron] by semigroupstechniques. Or use that its dual is compact:
L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω),
where ψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Compactness of L∗ from the Lp-maximal regularity, p < +∞:
‖∂tψ‖Lp(QT )+‖∆ψ‖Lp(QT ) ≤ C‖Θ‖Lp(QT ) ≤ C (T )‖Θ‖L∞(QT ).
For p large, compact embedding into L∞(QT ) !
Main ideas in the proof of the L1-THEOREM: first, anL1-compactness property
I The following mapping is compact :
L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
where w is the solution of∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I This is proved specifically in [Baras-Hassan-Veron] by semigroupstechniques. Or use that its dual is compact:
L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω),
where ψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Compactness of L∗ from the Lp-maximal regularity, p < +∞:
‖∂tψ‖Lp(QT )+‖∆ψ‖Lp(QT ) ≤ C‖Θ‖Lp(QT ) ≤ C (T )‖Θ‖L∞(QT ).
For p large, compact embedding into L∞(QT ) !
EXERCISE: Checking that L∗ is indeed the dual of LI L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
w solves :
∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I ?? L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω), whereψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Multiply the equation in w by ψ and integrate
−∫
Ωw0ψ(0)−
∫QT
w [∂tψ + d∆ψ] =
∫QT
g ψ,
I ⇒∫QT
w Θ =∫QT
g ψ +∫
Ω w0 ψ(0).
I < L(g ,w0),Θ >L1(QT )×L∞(QT )=< (g ,w0),L∗(Θ) >E1×E∞where Eq := Lq(QT )× Lq(Ω), q = 1 and +∞.
EXERCISE: Checking that L∗ is indeed the dual of LI L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
w solves :
∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I ?? L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω), whereψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Multiply the equation in w by ψ and integrate
−∫
Ωw0ψ(0)−
∫QT
w [∂tψ + d∆ψ] =
∫QT
g ψ,
I ⇒∫QT
w Θ =∫QT
g ψ +∫
Ω w0 ψ(0).
I < L(g ,w0),Θ >L1(QT )×L∞(QT )=< (g ,w0),L∗(Θ) >E1×E∞where Eq := Lq(QT )× Lq(Ω), q = 1 and +∞.
EXERCISE: Checking that L∗ is indeed the dual of LI L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
w solves :
∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I ?? L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω), whereψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Multiply the equation in w by ψ and integrate
−∫
Ωw0ψ(0)−
∫QT
w [∂tψ + d∆ψ] =
∫QT
g ψ,
I ⇒∫QT
w Θ =∫QT
g ψ +∫
Ω w0 ψ(0).
I < L(g ,w0),Θ >L1(QT )×L∞(QT )=< (g ,w0),L∗(Θ) >E1×E∞where Eq := Lq(QT )× Lq(Ω), q = 1 and +∞.
EXERCISE: Checking that L∗ is indeed the dual of LI L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
w solves :
∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I ?? L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω), whereψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Multiply the equation in w by ψ and integrate
−∫
Ωw0ψ(0)−
∫QT
w [∂tψ + d∆ψ] =
∫QT
g ψ,
I ⇒∫QT
w Θ =∫QT
g ψ +∫
Ω w0 ψ(0).
I < L(g ,w0),Θ >L1(QT )×L∞(QT )=< (g ,w0),L∗(Θ) >E1×E∞where Eq := Lq(QT )× Lq(Ω), q = 1 and +∞.
EXERCISE: Checking that L∗ is indeed the dual of LI L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )
w solves :
∂tw − d∆w = g in QT ,∂νw = 0 on ΣT , w(0, ·) = w0.
I ?? L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω), whereψ is the solution of the dual problem
–(∂tψ + d∆ψ) = Θ in QT , ∂νψ = 0 on ΣT , ψ(T ) = 0.
I Multiply the equation in w by ψ and integrate
−∫
Ωw0ψ(0)−
∫QT
w [∂tψ + d∆ψ] =
∫QT
g ψ,
I ⇒∫QT
w Θ =∫QT
g ψ +∫
Ω w0 ψ(0).
I < L(g ,w0),Θ >L1(QT )×L∞(QT )=< (g ,w0),L∗(Θ) >E1×E∞where Eq := Lq(QT )× Lq(Ω), q = 1 and +∞.
Back to the proof of the L1-THEOREMI We start from the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I By L1-compactness of the heat operator, there existsu ∈ [L1(QT )m]+, ∀T > 0 and a subsequence as n→ +∞
un → u in L1(QT )m and a.e., f ni (un)→ fi (u) a.e., i = 1, ...,m.
I The difficulty is to prove that f ni (un)→ fi (u) in L1(QT ) (??).
Note that fi (u) ∈ L1(QT ) by Fatou’s Lemma :∫QT
|fi (u)| ≤ lim infn→+∞
∫QT
|f ni (un)| < +∞.
Back to the proof of the L1-THEOREMI We start from the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I By L1-compactness of the heat operator, there existsu ∈ [L1(QT )m]+, ∀T > 0 and a subsequence as n→ +∞
un → u in L1(QT )m and a.e., f ni (un)→ fi (u) a.e., i = 1, ...,m.
I The difficulty is to prove that f ni (un)→ fi (u) in L1(QT ) (??).
Note that fi (u) ∈ L1(QT ) by Fatou’s Lemma :∫QT
|fi (u)| ≤ lim infn→+∞
∫QT
|f ni (un)| < +∞.
Back to the proof of the L1-THEOREMI We start from the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I By L1-compactness of the heat operator, there existsu ∈ [L1(QT )m]+, ∀T > 0 and a subsequence as n→ +∞
un → u in L1(QT )m and a.e., f ni (un)→ fi (u) a.e., i = 1, ...,m.
I The difficulty is to prove that f ni (un)→ fi (u) in L1(QT ) (??).
Note that fi (u) ∈ L1(QT ) by Fatou’s Lemma :∫QT
|fi (u)| ≤ lim infn→+∞
∫QT
|f ni (un)| < +∞.
Back to the proof of the L1-THEOREM
I We start from the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I The main part of the difficulty is to prove that u is asupersolution i.e.
∂tui − di∆ui ≥ fi (u) for i = 1, ...,m.
I Once this is done, it follows that u is a solution thanks to(M) !!
Back to the proof of the L1-THEOREM
I We start from the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I The main part of the difficulty is to prove that u is asupersolution i.e.
∂tui − di∆ui ≥ fi (u) for i = 1, ...,m.
I Once this is done, it follows that u is a solution thanks to(M) !!
Back to the proof of the L1-THEOREM
I We start from the approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I The main part of the difficulty is to prove that u is asupersolution i.e.
∂tui − di∆ui ≥ fi (u) for i = 1, ...,m.
I Once this is done, it follows that u is a solution thanks to(M) !!
”u supersolution” implies ”u solution” thanks to (M)I If we know that u is a supersolution, we may write
∂tui − di∆ui = fi (u) + µi , 0 ≤ µi (= nonnegative measure).
I On the other hand, for all n
∂t(∑i
uni )−∆(
∑i
diuni ) +
[−∑i
f ni (un)
]= 0,
where by (M), [...] ≥ 0, so that by Fatou’s Lemma
∀φ ∈ C∞0 (QT )+,
∫QT
φ[−∑i
fi (u)] ≤ lim infn→+∞
∫QT
φ[−∑i
f ni (un)].
I We deduce that, in the sense of distributions
∂t(∑i
ui )−∆(∑i
diui )−∑i
fi (u) ≤ 0.
I Coupling with the supersolution property where µi ≥ 0:∑i
(∂tui − di∆ui ) =∑i
fi (u) + µi ≤∑i
fi (u),
⇒∑i
µi ≤ 0⇒ [µi ≡ 0, ∀i ] ⇒ u is a solution.
”u supersolution” implies ”u solution” thanks to (M)I If we know that u is a supersolution, we may write
∂tui − di∆ui = fi (u) + µi , 0 ≤ µi (= nonnegative measure).
I On the other hand, for all n
∂t(∑i
uni )−∆(
∑i
diuni ) +
[−∑i
f ni (un)
]= 0,
where by (M), [...] ≥ 0, so that by Fatou’s Lemma
∀φ ∈ C∞0 (QT )+,
∫QT
φ[−∑i
fi (u)] ≤ lim infn→+∞
∫QT
φ[−∑i
f ni (un)].
I We deduce that, in the sense of distributions
∂t(∑i
ui )−∆(∑i
diui )−∑i
fi (u) ≤ 0.
I Coupling with the supersolution property where µi ≥ 0:∑i
(∂tui − di∆ui ) =∑i
fi (u) + µi ≤∑i
fi (u),
⇒∑i
µi ≤ 0⇒ [µi ≡ 0, ∀i ] ⇒ u is a solution.
”u supersolution” implies ”u solution” thanks to (M)I If we know that u is a supersolution, we may write
∂tui − di∆ui = fi (u) + µi , 0 ≤ µi (= nonnegative measure).
I On the other hand, for all n
∂t(∑i
uni )−∆(
∑i
diuni ) +
[−∑i
f ni (un)
]= 0,
where by (M), [...] ≥ 0, so that by Fatou’s Lemma
∀φ ∈ C∞0 (QT )+,
∫QT
φ[−∑i
fi (u)] ≤ lim infn→+∞
∫QT
φ[−∑i
f ni (un)].
I We deduce that, in the sense of distributions
∂t(∑i
ui )−∆(∑i
diui )−∑i
fi (u) ≤ 0.
I Coupling with the supersolution property where µi ≥ 0:∑i
(∂tui − di∆ui ) =∑i
fi (u) + µi ≤∑i
fi (u),
⇒∑i
µi ≤ 0⇒ [µi ≡ 0, ∀i ] ⇒ u is a solution.
”u supersolution” implies ”u solution” thanks to (M)I If we know that u is a supersolution, we may write
∂tui − di∆ui = fi (u) + µi , 0 ≤ µi (= nonnegative measure).
I On the other hand, for all n
∂t(∑i
uni )−∆(
∑i
diuni ) +
[−∑i
f ni (un)
]= 0,
where by (M), [...] ≥ 0, so that by Fatou’s Lemma
∀φ ∈ C∞0 (QT )+,
∫QT
φ[−∑i
fi (u)] ≤ lim infn→+∞
∫QT
φ[−∑i
f ni (un)].
I We deduce that, in the sense of distributions
∂t(∑i
ui )−∆(∑i
diui )−∑i
fi (u) ≤ 0.
I Coupling with the supersolution property where µi ≥ 0:∑i
(∂tui − di∆ui ) =∑i
fi (u) + µi ≤∑i
fi (u),
⇒∑i
µi ≤ 0⇒ [µi ≡ 0, ∀i ] ⇒ u is a solution.
L1-THEOREM: how to prove that u is a supersolution ?
I We use truncation techniques withr ∈ [0,+∞)→ Tk(r) = infr , k or a regularized version.
I An example of use of Tk for an equation. Suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,
Assume that supn∫
Ω |f (wn)| < +∞ and wn → w a.e. and inL1(QT ).
I EXERCICE: Prove that w is a super-solution, i.e.
∂tw − d∆w ≥ f (w).
I As noticed before we already know that f (w) ∈ L1(QT ) byFatou’s Lemma.
L1-THEOREM: how to prove that u is a supersolution ?
I We use truncation techniques withr ∈ [0,+∞)→ Tk(r) = infr , k or a regularized version.
I An example of use of Tk for an equation. Suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,
Assume that supn∫
Ω |f (wn)| < +∞ and wn → w a.e. and inL1(QT ).
I EXERCICE: Prove that w is a super-solution, i.e.
∂tw − d∆w ≥ f (w).
I As noticed before we already know that f (w) ∈ L1(QT ) byFatou’s Lemma.
L1-THEOREM: how to prove that u is a supersolution ?
I We use truncation techniques withr ∈ [0,+∞)→ Tk(r) = infr , k or a regularized version.
I An example of use of Tk for an equation. Suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,
Assume that supn∫
Ω |f (wn)| < +∞ and wn → w a.e. and inL1(QT ).
I EXERCICE: Prove that w is a super-solution, i.e.
∂tw − d∆w ≥ f (w).
I As noticed before we already know that f (w) ∈ L1(QT ) byFatou’s Lemma.
L1-THEOREM: how to prove that u is a supersolution ?
I We use truncation techniques withr ∈ [0,+∞)→ Tk(r) = infr , k or a regularized version.
I An example of use of Tk for an equation. Suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,
Assume that supn∫
Ω |f (wn)| < +∞ and wn → w a.e. and inL1(QT ).
I EXERCICE: Prove that w is a super-solution, i.e.
∂tw − d∆w ≥ f (w).
I As noticed before we already know that f (w) ∈ L1(QT ) byFatou’s Lemma.
L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, sup
n
∫Ω|f (wn)| < +∞.
and wn → w a.e. and in L1(QT ). Prove that
∂tw − d∆w ≥ f (w).
I Using the concavity of Tk ,
∂tTk(wn)−d∆Tk(wn) = T ′k(wn)[∂twn−d∆wn]−dT ′′k (wn)|∇wn|2,
⇒ ∂tTk(wn)− d∆Tk(wn)≥T ′k(wn)f (wn).
I wn → w a.e. ⇒ T ′k(wn)f (wn)→ T ′k(w)f (w) a.e..I [0 ≤ Tk ≤ k ; T ′k(r) = 0∀r ≥ k]⇒ by dominated convergence
Tk(wn)→ Tk(w) in L1(QT ), T ′k(wn)f (wn)→ T ′
k(w)f (w) in L1(QT ).
I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.
L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, sup
n
∫Ω|f (wn)| < +∞.
and wn → w a.e. and in L1(QT ). Prove that
∂tw − d∆w ≥ f (w).
I Using the concavity of Tk ,
∂tTk(wn)−d∆Tk(wn) = T ′k(wn)[∂twn−d∆wn]−dT ′′k (wn)|∇wn|2,
⇒ ∂tTk(wn)− d∆Tk(wn)≥T ′k(wn)f (wn).
I wn → w a.e. ⇒ T ′k(wn)f (wn)→ T ′k(w)f (w) a.e..I [0 ≤ Tk ≤ k ; T ′k(r) = 0∀r ≥ k]⇒ by dominated convergence
Tk(wn)→ Tk(w) in L1(QT ), T ′k(wn)f (wn)→ T ′
k(w)f (w) in L1(QT ).
I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.
L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, sup
n
∫Ω|f (wn)| < +∞.
and wn → w a.e. and in L1(QT ). Prove that
∂tw − d∆w ≥ f (w).
I Using the concavity of Tk ,
∂tTk(wn)−d∆Tk(wn) = T ′k(wn)[∂twn−d∆wn]−dT ′′k (wn)|∇wn|2,
⇒ ∂tTk(wn)− d∆Tk(wn)≥T ′k(wn)f (wn).
I wn → w a.e. ⇒ T ′k(wn)f (wn)→ T ′k(w)f (w) a.e..
I [0 ≤ Tk ≤ k ; T ′k(r) = 0∀r ≥ k]⇒ by dominated convergence
Tk(wn)→ Tk(w) in L1(QT ), T ′k(wn)f (wn)→ T ′
k(w)f (w) in L1(QT ).
I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.
L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, sup
n
∫Ω|f (wn)| < +∞.
and wn → w a.e. and in L1(QT ). Prove that
∂tw − d∆w ≥ f (w).
I Using the concavity of Tk ,
∂tTk(wn)−d∆Tk(wn) = T ′k(wn)[∂twn−d∆wn]−dT ′′k (wn)|∇wn|2,
⇒ ∂tTk(wn)− d∆Tk(wn)≥T ′k(wn)f (wn).
I wn → w a.e. ⇒ T ′k(wn)f (wn)→ T ′k(w)f (w) a.e..I [0 ≤ Tk ≤ k ; T ′k(r) = 0∀r ≥ k]⇒ by dominated convergence
Tk(wn)→ Tk(w) in L1(QT ), T ′k(wn)f (wn)→ T ′
k(w)f (w) in L1(QT ).
I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.
L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, sup
n
∫Ω|f (wn)| < +∞.
and wn → w a.e. and in L1(QT ). Prove that
∂tw − d∆w ≥ f (w).
I Using the concavity of Tk ,
∂tTk(wn)−d∆Tk(wn) = T ′k(wn)[∂twn−d∆wn]−dT ′′k (wn)|∇wn|2,
⇒ ∂tTk(wn)− d∆Tk(wn)≥T ′k(wn)f (wn).
I wn → w a.e. ⇒ T ′k(wn)f (wn)→ T ′k(w)f (w) a.e..I [0 ≤ Tk ≤ k ; T ′k(r) = 0∀r ≥ k]⇒ by dominated convergence
Tk(wn)→ Tk(w) in L1(QT ), T ′k(wn)f (wn)→ T ′
k(w)f (w) in L1(QT ).
I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,
I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.
L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have
∂twn − d∆wn = f (wn), wn ≥ 0, sup
n
∫Ω|f (wn)| < +∞.
and wn → w a.e. and in L1(QT ). Prove that
∂tw − d∆w ≥ f (w).
I Using the concavity of Tk ,
∂tTk(wn)−d∆Tk(wn) = T ′k(wn)[∂twn−d∆wn]−dT ′′k (wn)|∇wn|2,
⇒ ∂tTk(wn)− d∆Tk(wn)≥T ′k(wn)f (wn).
I wn → w a.e. ⇒ T ′k(wn)f (wn)→ T ′k(w)f (w) a.e..I [0 ≤ Tk ≤ k ; T ′k(r) = 0∀r ≥ k]⇒ by dominated convergence
Tk(wn)→ Tk(w) in L1(QT ), T ′k(wn)f (wn)→ T ′
k(w)f (w) in L1(QT ).
I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.
Using truncations in the proof of L1-THEOREM
I Approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I We introduce wni := Tk
(uni + η
∑j 6=i un
j
), i = 1, ...,m.
I ∂twni − di∆wn
i ≥ T ′k(wni )fi (un) + Rn
i (η, k)
I We prove that the limit u is a supersolution by successivelyletting n→ +∞, η → 0, k → +∞.
I Main estimate:∫
[uni ≤k] |∇uni |2 ≤ C k . !!
Using truncations in the proof of L1-THEOREM
I Approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I We introduce wni := Tk
(uni + η
∑j 6=i un
j
), i = 1, ...,m.
I ∂twni − di∆wn
i ≥ T ′k(wni )fi (un) + Rn
i (η, k)
I We prove that the limit u is a supersolution by successivelyletting n→ +∞, η → 0, k → +∞.
I Main estimate:∫
[uni ≤k] |∇uni |2 ≤ C k . !!
Using truncations in the proof of L1-THEOREM
I Approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I We introduce wni := Tk
(uni + η
∑j 6=i un
j
), i = 1, ...,m.
I ∂twni − di∆wn
i ≥ T ′k(wni )fi (un) + Rn
i (η, k)
I We prove that the limit u is a supersolution by successivelyletting n→ +∞, η → 0, k → +∞.
I Main estimate:∫
[uni ≤k] |∇uni |2 ≤ C k . !!
Using truncations in the proof of L1-THEOREM
I Approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I We introduce wni := Tk
(uni + η
∑j 6=i un
j
), i = 1, ...,m.
I ∂twni − di∆wn
i ≥ T ′k(wni )fi (un) + Rn
i (η, k)
I We prove that the limit u is a supersolution by successivelyletting n→ +∞, η → 0, k → +∞.
I Main estimate:∫
[uni ≤k] |∇uni |2 ≤ C k . !!
Using truncations in the proof of L1-THEOREM
I Approximate problem
(Sn∞)
∀i = 1, ...,m∂t u
ni − di ∆un
i = f ni (un) in Q∞,
∂νuni = 0 on Σ∞,
uni (0, ·) = infu0
i , n ≥ 0,
supn
∫QT
|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.
I We introduce wni := Tk
(uni + η
∑j 6=i un
j
), i = 1, ...,m.
I ∂twni − di∆wn
i ≥ T ′k(wni )fi (un) + Rn
i (η, k)
I We prove that the limit u is a supersolution by successivelyletting n→ +∞, η → 0, k → +∞.
I Main estimate:∫
[uni ≤k] |∇uni |2 ≤ C k . !!
EXERCICE: Prove∫
[uni ≤k] |∇u
ni |2 ≤ C k
I This is an L1-property of the heat equation∂tw − d∆w = f ∈ L1(QT )+,∂νw = 0, w(0) = w0 ∈ L1(Ω)+,w ≥ 0.
I Claim: d∫
[w≤k] |∇w |2 ≤ k[∫
QTf +
∫Ω w0
].
I Multiply the equation in w by Tk(w). Let jk(r) =∫ r
0 Tk(s)ds.Integrating by parts and using−∫QT
Tk(w)∆w =∫QT∇[Tk(w)]∇w =
∫QT
T ′k(w)|∇w |2 imply∫
QT
Tk(w)∂tw + d
∫QT
T ′k(w)|∇w |2 =
∫QT
Tk(w)f .
I Recall Tk(r) = infr , k,Tk(r) ≤ k,T ′k(r) = χ[r≤k]. ∫Ω jk(w(T )) + d
∫[w≤k] |∇w |2 =
∫QT
Tk(w)f +∫
Ω jk(w0)
≤ k[∫
QTf +
∫Ω w0
].
I Prove the version on [|w | ≤ k] when f ,w0 do not have any sign.
EXERCICE: Prove∫
[uni ≤k] |∇u
ni |2 ≤ C k
I This is an L1-property of the heat equation∂tw − d∆w = f ∈ L1(QT )+,∂νw = 0, w(0) = w0 ∈ L1(Ω)+,w ≥ 0.
I Claim: d∫
[w≤k] |∇w |2 ≤ k[∫
QTf +
∫Ω w0
].
I Multiply the equation in w by Tk(w). Let jk(r) =∫ r
0 Tk(s)ds.Integrating by parts and using−∫QT
Tk(w)∆w =∫QT∇[Tk(w)]∇w =
∫QT
T ′k(w)|∇w |2 imply∫
QT
Tk(w)∂tw + d
∫QT
T ′k(w)|∇w |2 =
∫QT
Tk(w)f .
I Recall Tk(r) = infr , k,Tk(r) ≤ k,T ′k(r) = χ[r≤k]. ∫Ω jk(w(T )) + d
∫[w≤k] |∇w |2 =
∫QT
Tk(w)f +∫
Ω jk(w0)
≤ k[∫
QTf +
∫Ω w0
].
I Prove the version on [|w | ≤ k] when f ,w0 do not have any sign.
EXERCICE: Prove∫
[uni ≤k] |∇u
ni |2 ≤ C k
I This is an L1-property of the heat equation∂tw − d∆w = f ∈ L1(QT )+,∂νw = 0, w(0) = w0 ∈ L1(Ω)+,w ≥ 0.
I Claim: d∫
[w≤k] |∇w |2 ≤ k[∫
QTf +
∫Ω w0
].
I Multiply the equation in w by Tk(w). Let jk(r) =∫ r
0 Tk(s)ds.Integrating by parts and using−∫QT
Tk(w)∆w =∫QT∇[Tk(w)]∇w =
∫QT
T ′k(w)|∇w |2 imply∫
QT
Tk(w)∂tw + d
∫QT
T ′k(w)|∇w |2 =
∫QT
Tk(w)f .
I Recall Tk(r) = infr , k,Tk(r) ≤ k,T ′k(r) = χ[r≤k]. ∫Ω jk(w(T )) + d
∫[w≤k] |∇w |2 =
∫QT
Tk(w)f +∫
Ω jk(w0)
≤ k[∫
QTf +
∫Ω w0
].
I Prove the version on [|w | ≤ k] when f ,w0 do not have any sign.
EXERCICE: Prove∫
[uni ≤k] |∇u
ni |2 ≤ C k
I This is an L1-property of the heat equation∂tw − d∆w = f ∈ L1(QT )+,∂νw = 0, w(0) = w0 ∈ L1(Ω)+,w ≥ 0.
I Claim: d∫
[w≤k] |∇w |2 ≤ k[∫
QTf +
∫Ω w0
].
I Multiply the equation in w by Tk(w). Let jk(r) =∫ r
0 Tk(s)ds.Integrating by parts and using−∫QT
Tk(w)∆w =∫QT∇[Tk(w)]∇w =
∫QT
T ′k(w)|∇w |2 imply∫
QT
Tk(w)∂tw + d
∫QT
T ′k(w)|∇w |2 =
∫QT
Tk(w)f .
I Recall Tk(r) = infr , k,Tk(r) ≤ k,T ′k(r) = χ[r≤k]. ∫Ω jk(w(T )) + d
∫[w≤k] |∇w |2 =
∫QT
Tk(w)f +∫
Ω jk(w0)
≤ k[∫
QTf +
∫Ω w0
].
I Prove the version on [|w | ≤ k] when f ,w0 do not have any sign.
EXERCICE: Prove∫
[uni ≤k] |∇u
ni |2 ≤ C k
I This is an L1-property of the heat equation∂tw − d∆w = f ∈ L1(QT )+,∂νw = 0, w(0) = w0 ∈ L1(Ω)+,w ≥ 0.
I Claim: d∫
[w≤k] |∇w |2 ≤ k[∫
QTf +
∫Ω w0
].
I Multiply the equation in w by Tk(w). Let jk(r) =∫ r
0 Tk(s)ds.Integrating by parts and using−∫QT
Tk(w)∆w =∫QT∇[Tk(w)]∇w =
∫QT
T ′k(w)|∇w |2 imply∫
QT
Tk(w)∂tw + d
∫QT
T ′k(w)|∇w |2 =
∫QT
Tk(w)f .
I Recall Tk(r) = infr , k,Tk(r) ≤ k,T ′k(r) = χ[r≤k]. ∫Ω jk(w(T )) + d
∫[w≤k] |∇w |2 =
∫QT
Tk(w)f +∫
Ω jk(w0)
≤ k[∫
QTf +
∫Ω w0
].
I Prove the version on [|w | ≤ k] when f ,w0 do not have any sign.
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I We already saw the case whenf1(u1, u2) = −uα1 h(u2),f2(u1, u2) = uα1 h(u2),
(1)
where h ≥ 0 with any growth at infinity and α ∈ [1,+∞).Due to the ”cheap” estimate∫
QT
uα1 h(u2) ≤∫
Ωu0
1 .
I Assume m = 2, (P)+(M) and also
(Mλ) f1(u) + λf2(u) ≤ 0 for some λ 6= 1.
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I We already saw the case whenf1(u1, u2) = −uα1 h(u2),f2(u1, u2) = uα1 h(u2),
(1)
where h ≥ 0 with any growth at infinity and α ∈ [1,+∞).Due to the ”cheap” estimate∫
QT
uα1 h(u2) ≤∫
Ωu0
1 .
I Assume m = 2, (P)+(M) and also
(Mλ) f1(u) + λf2(u) ≤ 0 for some λ 6= 1.
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I Assume m = 2, (P)+(M) and also
(Mλ) f1(u) + λf2(u) ≤ 0 for some λ 6= 1.
I Integrate the equation
∂t(u1 + λu2)−∆(d1u1 + λd2u2)− [f1(u) + λf2(u)] = 0,
⇒∫
Ω(u1 +λu2)(T ) + 0 +
∫QT
|f1(u) + λf2(u)| =
∫Ω
u01 +λu0
2 .
I This for some λ 6= 1 and also for λ = 1 (by (M)). Thus∫QT
|f1(u)|+ |f2(u)| ≤ C
(∫Ω
u01 + u0
2
).
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I Assume m = 2, (P)+(M) and also
(Mλ) f1(u) + λf2(u) ≤ 0 for some λ 6= 1.
I Integrate the equation
∂t(u1 + λu2)−∆(d1u1 + λd2u2)− [f1(u) + λf2(u)] = 0,
⇒∫
Ω(u1 +λu2)(T ) + 0 +
∫QT
|f1(u) + λf2(u)| =
∫Ω
u01 +λu0
2 .
I This for some λ 6= 1 and also for λ = 1 (by (M)). Thus∫QT
|f1(u)|+ |f2(u)| ≤ C
(∫Ω
u01 + u0
2
).
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I Assume m = 2, (P)+(M) and also
(Mλ) f1(u) + λf2(u) ≤ 0 for some λ 6= 1.
I Integrate the equation
∂t(u1 + λu2)−∆(d1u1 + λd2u2)− [f1(u) + λf2(u)] = 0,
⇒∫
Ω(u1 +λu2)(T ) + 0 +
∫QT
|f1(u) + λf2(u)| =
∫Ω
u01 +λu0
2 .
I This for some λ 6= 1 and also for λ = 1 (by (M)). Thus∫QT
|f1(u)|+ |f2(u)| ≤ C
(∫Ω
u01 + u0
2
).
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I The 2× 2 L∞(Ω)-blow up examples given previously didsatisfy (Mλ) for λ close to 1. This is why, althoughu1(t), u2(t) blow up in finite time, they may be extended asglobal weak solutions on [0,+∞).
I Another example isf1(u) = λup1
1 up22 − uq1
1 uq22 ,
f2(u) = uq11 uq2
2 − up11 up2
2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).
I Then global existence of weak solutions sincef1 + f2 = (λ− 1)up1
1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1
1 uq22 ≤ 0.
I Open for λ = 1 (except for some values of pi , qi ).
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I The 2× 2 L∞(Ω)-blow up examples given previously didsatisfy (Mλ) for λ close to 1. This is why, althoughu1(t), u2(t) blow up in finite time, they may be extended asglobal weak solutions on [0,+∞).
I Another example isf1(u) = λup1
1 up22 − uq1
1 uq22 ,
f2(u) = uq11 uq2
2 − up11 up2
2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).
I Then global existence of weak solutions sincef1 + f2 = (λ− 1)up1
1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1
1 uq22 ≤ 0.
I Open for λ = 1 (except for some values of pi , qi ).
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I The 2× 2 L∞(Ω)-blow up examples given previously didsatisfy (Mλ) for λ close to 1. This is why, althoughu1(t), u2(t) blow up in finite time, they may be extended asglobal weak solutions on [0,+∞).
I Another example isf1(u) = λup1
1 up22 − uq1
1 uq22 ,
f2(u) = uq11 uq2
2 − up11 up2
2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).
I Then global existence of weak solutions sincef1 + f2 = (λ− 1)up1
1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1
1 uq22 ≤ 0.
I Open for λ = 1 (except for some values of pi , qi ).
L1-THEOREM applies to many situations
(S∞)
∀i = 1, ...,m∂tui − di ∆ui = fi (u1, u2, ..., um) in Q∞,∂νui = 0 on Σ∞,ui (0, ·) = u0
i (·) ≥ 0.
I The 2× 2 L∞(Ω)-blow up examples given previously didsatisfy (Mλ) for λ close to 1. This is why, althoughu1(t), u2(t) blow up in finite time, they may be extended asglobal weak solutions on [0,+∞).
I Another example isf1(u) = λup1
1 up22 − uq1
1 uq22 ,
f2(u) = uq11 uq2
2 − up11 up2
2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).
I Then global existence of weak solutions sincef1 + f2 = (λ− 1)up1
1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1
1 uq22 ≤ 0.
I Open for λ = 1 (except for some values of pi , qi ).
L1-THEOREM applies to many situations
I More generally it applies if there exists an invertible matrix Qwith nonnegative entries such that
∀r ∈ [0,∞)m, Q f (r) ≤ [1 +∑
1≤i≤mri ]b,
for some b ∈ IRm, f = (f1, ..., fm)t .(In other words, if there are m linearly independentinequalities for the fi ’s and not only one).
I By inverting Q, we prove the a priori L1(QT )- estimates for allfi (u). Whence global existence of weak solutions.
I This L1-THEOREM extends to RD-systems with nonlineardegenerate diffusions of the kind −∆umi . See the talk by ElHaj Laamri tomorrow.
L1-THEOREM applies to many situations
I More generally it applies if there exists an invertible matrix Qwith nonnegative entries such that
∀r ∈ [0,∞)m, Q f (r) ≤ [1 +∑
1≤i≤mri ]b,
for some b ∈ IRm, f = (f1, ..., fm)t .(In other words, if there are m linearly independentinequalities for the fi ’s and not only one).
I By inverting Q, we prove the a priori L1(QT )- estimates for allfi (u). Whence global existence of weak solutions.
I This L1-THEOREM extends to RD-systems with nonlineardegenerate diffusions of the kind −∆umi . See the talk by ElHaj Laamri tomorrow.
L1-THEOREM applies to many situations
I More generally it applies if there exists an invertible matrix Qwith nonnegative entries such that
∀r ∈ [0,∞)m, Q f (r) ≤ [1 +∑
1≤i≤mri ]b,
for some b ∈ IRm, f = (f1, ..., fm)t .(In other words, if there are m linearly independentinequalities for the fi ’s and not only one).
I By inverting Q, we prove the a priori L1(QT )- estimates for allfi (u). Whence global existence of weak solutions.
I This L1-THEOREM extends to RD-systems with nonlineardegenerate diffusions of the kind −∆umi . See the talk by ElHaj Laamri tomorrow.