Existence globale pour des systèmes de réaction-diffusion ...

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.

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 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,

∀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

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

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 For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× Ω

i (·) ≥ 0.

I (M):∑

1≤i≤m fi(r) ≤ 0, ∀ r = (r1, ..., rm) ∈ [0,+∞)m.

I For T ∈ (0,+∞], QT := (0,T )× Ω, ΣT := (0,T )× Ω

i (·) ≥ 0.

I (M):∑

1≤i≤m fi(r) ≤ 0, ∀ r = (r1, ..., rm) ∈ [0,+∞)m.

I About the condition (P):

∀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.

f1(0, r2)≥ 0

f2(0, r2)

AAAKf1(r1, 0)

f2(r1, 0)≥ 0

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 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 =

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

u−i fi (u) = 2

u−i fi (u+)≥ 0 ,

I ⇒ ∂t∫

Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0.

+1 , ..., r

+i−1, 0, r

+i+1, ..., r

+m ) ≥ 0, ∀r ∈ IRm

u−i fi (u).

−∫

Ω∂t(u−i )2+di |∇(u−i )2| = 2

u−i fi (u) = 2

u−i fi (u+)≥ 0 ,

I ⇒ ∂t∫

Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0.

+1 , ..., r

+i−1, 0, r

+i+1, ..., r

+m ) ≥ 0, ∀r ∈ IRm

u−i fi (u).

−∫

Ω∂t(u−i )2+di |∇(u−i )2| = 2

u−i fi (u) = 2

u−i fi (u+)≥ 0 ,

I ⇒ ∂t∫

Ω(u−i )2 ≤ 0 ⇒ (u−i )2(t) ≡ 0, ∀t ≥ 0.

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0.

+1 , ..., r

+i−1, 0, r

+i+1, ..., r

+m ) ≥ 0, ∀r ∈ IRm

u−i fi (u).

−∫

Ω∂t(u−i )2+di |∇(u−i )2| = 2

u−i fi (u) = 2

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’:∫

∑1≤i≤m

ui (t, x)dx ≤∫

∑1≤i≤m

u0i (x)dx , ∀t ≥ 0.

Add up the m equations, integrate on Ω, use∫

Ω∆ui =

∫∂Ω∂νui = 0:∫

∂t [∑

ui (t)]dx =

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’:∫

∑1≤i≤m

ui (t, x)dx ≤∫

∑1≤i≤m

u0i (x)dx , ∀t ≥ 0.

Add up the m equations, integrate on Ω, use∫

Ω∆ui =

∫∂Ω∂νui = 0:∫

∂t [∑

ui (t)]dx =

fi (u)dx ≤ 0.

I ⇒ L1(Ω)- a priori estimates, uniform in time (t ∈ [0,T ∗) ).

Same with (P)+(M’).

∂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) ).

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0.

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 < +∞

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0.

I (M’)∑

1≤i≤m ai fi (r) ≤ K0 + K1∑

I ⇒ ‖ui (t)‖L1(Ω) ≤ C(T , ‖u0‖L1(Ω)m

), ∀t ∈ [0,T ],∀T < +∞

∂νui = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0.

I (M’)∑

1≤i≤m ai fi (r) ≤ K0 + K1∑

I ⇒ ‖ui (t)‖L1(Ω) ≤ C(T , ‖u0‖L1(Ω)m

), ∀t ∈ [0,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 ∗ = +∞.

(ODE )

∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0

i ∈ [0,+∞) ???

]⇒ [T ∗ +∞].

I (M) ⇒ ∂t (∑

i ui (t)) =∑

i fi (u) ≤ 0.

I ⇒∑

i ui (t) ≤∑

i u0i , ∀t ∈ [0,T ∗)

(ODE )

∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0

i ∈ [0,+∞) ???

]⇒ [T ∗ +∞].

I (M) ⇒ ∂t (∑

i ui (t)) =∑

i fi (u) ≤ 0.

I ⇒∑

i ui (t) ≤∑

i u0i , ∀t ∈ [0,T ∗)

(ODE )

∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0

i ∈ [0,+∞) ???

]⇒ [T ∗ +∞].

I (M) ⇒ ∂t (∑

i ui (t)) =∑

i fi (u) ≤ 0.

I ⇒∑

i ui (t) ≤∑

i u0i , ∀t ∈ [0,T ∗)

(ODE )

∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0

i ∈ [0,+∞) ???

]⇒ [T ∗ +∞].

I (M) ⇒ ∂t (∑

i ui (t)) =∑

i fi (u) ≤ 0.

I ⇒∑

i ui (t) ≤∑

i u0i , ∀t ∈ [0,T ∗)

(ODE )

∀i = 1, ...,m,∂tui = fi (u1, ..., um),ui (0) = u0

i ∈ [0,+∞) ???

]⇒ [T ∗ +∞].

I (M) ⇒ ∂t (∑

i ui (t)) =∑

i fi (u) ≤ 0.

I ⇒∑

i ui (t) ≤∑

i u0i , ∀t ∈ [0,T ∗)

Local existence holds for the full RD-system

∀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.

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

supt∈[0,T∗)

‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,

]⇒ [T ∗ = +∞]

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

supt∈[0,T∗)

‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,

]⇒ [T ∗ = +∞]

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

supt∈[0,T∗)

‖ui(t)‖L∞(Ω) < +∞, ∀ 1 ≤ i ≤ m,

]⇒ [T ∗ = +∞]

EXERCISE: Global existence when di = d ,∀i

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

)− d∆

)=∑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 ∗ = +∞.

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

)− d∆

)=∑i

fi(u)≤ 0.

‖∑i

ui(t)‖L∞(Ω) ≤ ‖∑i

u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).

⇒ T ∗ = +∞.

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

)− d∆

)=∑i

fi(u)≤ 0.

‖∑i

ui(t)‖L∞(Ω) ≤ ‖∑i

u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).

⇒ T ∗ = +∞.

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

)− d∆

)=∑i

fi(u)≤ 0.

‖∑i

ui(t)‖L∞(Ω) ≤ ‖∑i

u0i ‖L∞(Ω), ∀t ∈ [0,T ∗).

⇒ 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 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.

ui (0, ·) = u0i ∈ C∞(Ω)+,

with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.

ui (0, ·) = u0i ∈ C∞(Ω)+,

with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.

ui (0, ·) = u0i ∈ C∞(Ω)+,

with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.

ui (0, ·) = u0i ∈ C∞(Ω)+,

with c1(t, x) + c2(t, x) ≤ 0, α + β = 2 + η.

Finite time blow up with (P)+(M): ideas of the proof

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.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.

ui (0, ·) = u0i ∈ C∞(Ω)+,

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.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

∂tu1 − d1∆u1 + ∂tu2 − d2∆u2 ≤ 0.

ui (0, ·) = u0i ∈ C∞(Ω)+,

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 !

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

ui (0, ·) = u0i ∈ C∞(Ω)+,

ui (t, x) =ai (T − t) + bi |x |2

(T − t + |x |2)γ, ai , bi ∈ (0,+∞), i = 1, 2, γ > 1.

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 ).

i ≥ 0, i = 1, 2.

‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).

u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).

i ≥ 0, i = 1, 2.

‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).

u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).

i ≥ 0, i = 1, 2.

‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).

u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).

i ≥ 0, i = 1, 2.

‖u1(t)‖L∞(Ω) ≤ ‖u01‖L∞(Ω).

u2 = −A(u1), A := (∂t − d2∆)−1(∂t − d1∆).

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.

∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0

i ≥ 0, i = 1, 2.

∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0

i ≥ 0, i = 1, 2.

∂νu1 = 0 = ∂νu2 on ΣT ,ui (0, ·) = u0

i ≥ 0, i = 1, 2.

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.

(Ineq)

(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].

(Ineq)

(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].

(Ineq)

(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].

for some C = C (q, d2,Ω) independent of T .

I The expected estimate (∗) is exactly the dual of (∗∗) with q = p′.

(Ineq)

(∗) ‖u2‖Lp(QT ) ≤ C [T 1/p + ‖u1‖Lp(QT )].

The proof of the Lp-estimate

(Ineq)

I Multiplying (Ineq) by ψ ≥ 0 and integrating by parts give

−∫QT

u2(∂tψ + d2∆ψ) ≤∫

[u02 − a u0

1 ]ψ(0) +

u1[−a∂tψ + b∆ψ],

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 .

(Ineq)

−∫QT

u2(∂tψ + d2∆ψ) ≤∫

[u02 − a u0

1 ]ψ(0) +

u1[−a∂tψ + b∆ψ],

u2Θ ≤ ‖u02 − au0

1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)

+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).

I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T

I∫QT

u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C

]‖Θ‖Lp′ (QT ),

I Whence by duality

(Ineq)

−∫QT

u2(∂tψ + d2∆ψ) ≤∫

[u02 − a u0

1 ]ψ(0) +

u1[−a∂tψ + b∆ψ],

u2Θ ≤ ‖u02 − au0

1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)

+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).

I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T

I∫QT

u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C

]‖Θ‖Lp′ (QT ),

I Whence by duality

(Ineq)

−∫QT

u2(∂tψ + d2∆ψ) ≤∫

[u02 − a u0

1 ]ψ(0) +

u1[−a∂tψ + b∆ψ],

u2Θ ≤ ‖u02 − au0

1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)

+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).

I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T

I∫QT

u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C

]‖Θ‖Lp′ (QT ),

I Whence by duality

(Ineq)

−∫QT

u2(∂tψ + d2∆ψ) ≤∫

[u02 − a u0

1 ]ψ(0) +

u1[−a∂tψ + b∆ψ],

u2Θ ≤ ‖u02 − au0

1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)

+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).

I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T

I∫QT

u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C

]‖Θ‖Lp′ (QT ),

I Whence by duality

(Ineq)

−∫QT

u2(∂tψ + d2∆ψ) ≤∫

[u02 − a u0

1 ]ψ(0) +

u1[−a∂tψ + b∆ψ],

u2Θ ≤ ‖u02 − au0

1‖Lp(Ω)‖ψ(0)‖Lp′ (Ω)

+‖u1‖Lp(QT )‖ − a∂tψ + b∆ψ‖Lp′ (QT ).

I ‖ψ(0)‖Lp′ (Ω) = ‖∫ T

I∫QT

u2Θ ≤[CT 1/pC0 + ‖u1‖Lp(QT )(|a|+ |b|)C

]‖Θ‖Lp′ (QT ),

I Whence by duality

Two main applications of the Lp approach

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].

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

Q f (r) ≤ [1 +∑i

and with an at most polynomial growth for f .

I Then global existence of classical solutions holds for thecorresponding RD-system.

[1 01 1

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

Q f (r) ≤ [1 +∑i

[1 01 1

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

Q f (r) ≤ [1 +∑i

[1 01 1

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

Q f (r) ≤ [1 +∑i

[1 01 1

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 ≤ ∆

(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 )].

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

∂t(∑i

ui )−d∆(∑i

ui ) =∑i

(di−d)∆ui+∑i

fi ≤ ∆

(di − d)ui

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

∂t(∑i

ui )−d∆(∑i

ui ) =∑i

(di−d)∆ui+∑i

fi ≤ ∆

(di − d)ui

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

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.

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

I ⇒ ‖∑

i ui‖Lp(QT )].

I ⇒ ‖∑

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

I ⇒ ‖∑

i ui‖Lp(QT )].

I ⇒ ‖∑

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

I ⇒ ‖∑

i ui‖Lp(QT )].

I ⇒ ‖∑

i (·) ≥ 0, u0i ∈ L∞(Ω)+.

‖∑i

ui‖Lp(QT ) ≤ C [T 1/p + ‖∑i

I ⇒ ‖∑

i ui‖Lp(QT )].

I ⇒ ‖∑

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 ).

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.

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.

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:∫

|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 +

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 (·) ≥ 0.

I L1-THEOREM. Assume that (P)+(M) or +(M’) hold.Assume moreover that the following L1-a priori estimate holds:∫

|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 +

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 (·) ≥ 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

|f ni (un)| < +∞, ∀T > 0, ∀i = 1, ...,m.

(S∞)

i (·) ≥ 0.

|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m ?

1+n−1∑

j |fj (r)| , i = 1, ...,m. Then

i in (S∞), has a global classical solution un.

I Saying that the L1-a priori estimate holds means that

|f ni (un)| < +∞, ∀T > 0, ∀i = 1, ...,m.

(S∞)

i (·) ≥ 0.

|fi (u)| ≤ C < +∞, ∀T > 0, ∀i = 1, ...,m ?

1+n−1∑

j |fj (r)| , i = 1, ...,m. Then

i in (S∞), has a global classical solution un.I Saying that the L1-a priori estimate holds means that

|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 +

uα1 h(u2) =

I This implies the a priori L1-bound on the nonlinear reaction:∫QT

uα1 h(u2) ≤∫

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.

i ≥ 0, i = 1, 2.

q, q ≥ 2.

u1(T )− 0 +

uα1 h(u2) =

uα1 h(u2) ≤∫

i ≥ 0, i = 1, 2.

q, q ≥ 2.

u1(T )− 0 +

uα1 h(u2) =

uα1 h(u2) ≤∫

i ≥ 0, i = 1, 2.

q, q ≥ 2.

u1(T )− 0 +

uα1 h(u2) =

uα1 h(u2) ≤∫

i ≥ 0, i = 1, 2.

q, q ≥ 2.

u1(T )− 0 +

uα1 h(u2) =

uα1 h(u2) ≤∫

i ≥ 0, i = 1, 2.

q, q ≥ 2.

u1(T )− 0 +

uα1 h(u2) =

uα1 h(u2) ≤∫

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

|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

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

|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 ) !

L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )

L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω),

L : (g ,w0) ∈ L1(QT )× L1(Ω) 7→ w ∈ L1(QT )

L∗ : Θ ∈ L∞(QT ) 7→ (ψ,ψ(0)) ∈ L∞(QT )× L∞(Ω),

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

I Multiply the equation in w by ψ and integrate

−∫

Ωw0ψ(0)−

w [∂tψ + d∆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 +∞.

w solves :

−∫

Ωw0ψ(0)−

w [∂tψ + d∆ψ] =

I ⇒∫QT

w Θ =∫QT

g ψ +∫

Ω w0 ψ(0).

w solves :

−∫

Ωw0ψ(0)−

w [∂tψ + d∆ψ] =

I ⇒∫QT

w Θ =∫QT

g ψ +∫

Ω w0 ψ(0).

w solves :

−∫

Ωw0ψ(0)−

w [∂tψ + d∆ψ] =

I ⇒∫QT

w Θ =∫QT

g ψ +∫

Ω w0 ψ(0).

w solves :

−∫

Ωw0ψ(0)−

w [∂tψ + d∆ψ] =

I ⇒∫QT

w Θ =∫QT

g ψ +∫

Ω w0 ψ(0).

Back to the proof of the L1-THEOREMI We start from the approximate problem

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|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→+∞

|f ni (un)| < +∞.

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

|f ni (un)| < +∞.

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

|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

uni (0, ·) = infu0

i , n ≥ 0,

|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) !!

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,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 )−∆(

diuni ) +

[−∑i

f ni (un)

where by (M), [...] ≥ 0, so that by Fatou’s Lemma

∀φ ∈ C∞0 (QT )+,

φ[−∑i

fi (u)] ≤ lim infn→+∞

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.

∂t(∑i

uni )−∆(

diuni ) +

[−∑i

f ni (un)

∀φ ∈ C∞0 (QT )+,

φ[−∑i

f ni (un)].

∂t(∑i

ui )−∆(∑i

diui )−∑i

fi (u) ≤ 0.

fi (u),

⇒∑i

∂t(∑i

uni )−∆(

diuni ) +

[−∑i

f ni (un)

∀φ ∈ C∞0 (QT )+,

φ[−∑i

f ni (un)].

∂t(∑i

ui )−∆(∑i

diui )−∑i

fi (u) ≤ 0.

fi (u),

⇒∑i

∂t(∑i

uni )−∆(

diuni ) +

[−∑i

f ni (un)

∀φ ∈ C∞0 (QT )+,

φ[−∑i

f ni (un)].

∂t(∑i

ui )−∆(∑i

diui )−∑i

fi (u) ≤ 0.

fi (u),

⇒∑i

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.

∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,

Assume that supn∫

∂tw − d∆w ≥ f (w).

∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,

Assume that supn∫

∂tw − d∆w ≥ f (w).

∂twn − d∆wn = f (wn), wn ≥ 0, ∀n,

Assume that supn∫

∂tw − d∆w ≥ f (w).

L1-THEOREM: how to prove that u is a supersolution ?I EXERCISE: suppose we have

∂twn − d∆wn = f (wn), wn ≥ 0, sup

∫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 !.

∫Ω|f (wn)| < +∞.

∂tw − d∆w ≥ f (w).

∫Ω|f (wn)| < +∞.

∂tw − d∆w ≥ f (w).

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

∫Ω|f (wn)| < +∞.

∂tw − d∆w ≥ f (w).

∫Ω|f (wn)| < +∞.

∂tw − d∆w ≥ f (w).

I ⇒ ∂tTk(w)− d∆Tk(w) ≥ T ′k(w)f (w), ∀k ,

I ⇒ ∂tw − d∆w ≥ f (w) ⇒ w is a supersolution !.

∫Ω|f (wn)| < +∞.

∂tw − d∆w ≥ f (w).

Using truncations in the proof of L1-THEOREM

I Approximate problem

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

I We introduce wni := Tk

(uni + η

∑j 6=i un

), 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 . !!

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

(uni + η

∑j 6=i un

), i = 1, ...,m.

i (η, k)

I Main estimate:∫

[uni ≤k] |∇uni |2 ≤ C k . !!

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

(uni + η

∑j 6=i un

), i = 1, ...,m.

i (η, k)

I Main estimate:∫

[uni ≤k] |∇uni |2 ≤ C k . !!

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

(uni + η

∑j 6=i un

), i = 1, ...,m.

i (η, k)

I Main estimate:∫

[uni ≤k] |∇uni |2 ≤ C k . !!

(Sn∞)

∀i = 1, ...,m∂t u

ni − di ∆un

uni (0, ·) = infu0

i , n ≥ 0,

|f ni (un)| < +∞, ∀T > 0, for i = 1, ...,m.

(uni + η

∑j 6=i un

), i = 1, ...,m.

i (η, 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[∫

∫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 =

T ′k(w)|∇w |2 imply∫

Tk(w)∂tw + d

T ′k(w)|∇w |2 =

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 =

Tk(w)f +∫

Ω jk(w0)

≤ k[∫

∫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 Claim: d∫

[w≤k] |∇w |2 ≤ k[∫

∫Ω w0

Tk(w)∂tw + d

T ′k(w)|∇w |2 =

Tk(w)f .

∫[w≤k] |∇w |2 =

Tk(w)f +∫

Ω jk(w0)

≤ k[∫

∫Ω w0

EXERCICE: Prove∫

[uni ≤k] |∇u

ni |2 ≤ C k

I Claim: d∫

[w≤k] |∇w |2 ≤ k[∫

∫Ω w0

Tk(w)∂tw + d

T ′k(w)|∇w |2 =

Tk(w)f .

∫[w≤k] |∇w |2 =

Tk(w)f +∫

Ω jk(w0)

≤ k[∫

∫Ω w0

EXERCICE: Prove∫

[uni ≤k] |∇u

ni |2 ≤ C k

I Claim: d∫

[w≤k] |∇w |2 ≤ k[∫

∫Ω w0

Tk(w)∂tw + d

T ′k(w)|∇w |2 =

Tk(w)f .

∫[w≤k] |∇w |2 =

Tk(w)f +∫

Ω jk(w0)

≤ k[∫

∫Ω w0

EXERCICE: Prove∫

[uni ≤k] |∇u

ni |2 ≤ C k

I Claim: d∫

[w≤k] |∇w |2 ≤ k[∫

∫Ω w0

Tk(w)∂tw + d

T ′k(w)|∇w |2 =

Tk(w)f .

∫[w≤k] |∇w |2 =

Tk(w)f +∫

Ω jk(w0)

≤ k[∫

∫Ω w0

L1-THEOREM applies to many situations

(S∞)

i (·) ≥ 0.

I We already saw the case whenf1(u1, u2) = −uα1 h(u2),f2(u1, u2) = uα1 h(u2),

where h ≥ 0 with any growth at infinity and α ∈ [1,+∞).Due to the ”cheap” estimate∫

uα1 h(u2) ≤∫

I Assume m = 2, (P)+(M) and also

(Mλ) f1(u) + λf2(u) ≤ 0 for some λ 6= 1.

(S∞)

i (·) ≥ 0.

I We already saw the case whenf1(u1, u2) = −uα1 h(u2),f2(u1, u2) = uα1 h(u2),

where h ≥ 0 with any growth at infinity and α ∈ [1,+∞).Due to the ”cheap” estimate∫

uα1 h(u2) ≤∫

(S∞)

i (·) ≥ 0.

I Integrate the equation

∂t(u1 + λu2)−∆(d1u1 + λd2u2)− [f1(u) + λf2(u)] = 0,

⇒∫

Ω(u1 +λu2)(T ) + 0 +

|f1(u) + λf2(u)| =

u01 +λu0

I This for some λ 6= 1 and also for λ = 1 (by (M)). Thus∫QT

|f1(u)|+ |f2(u)| ≤ C

(∫Ω

u01 + u0

(S∞)

i (·) ≥ 0.

∂t(u1 + λu2)−∆(d1u1 + λd2u2)− [f1(u) + λf2(u)] = 0,

⇒∫

Ω(u1 +λu2)(T ) + 0 +

|f1(u) + λf2(u)| =

u01 +λu0

|f1(u)|+ |f2(u)| ≤ C

(∫Ω

u01 + u0

(S∞)

i (·) ≥ 0.

∂t(u1 + λu2)−∆(d1u1 + λd2u2)− [f1(u) + λf2(u)] = 0,

⇒∫

Ω(u1 +λu2)(T ) + 0 +

|f1(u) + λf2(u)| =

u01 +λu0

|f1(u)|+ |f2(u)| ≤ C

(∫Ω

u01 + u0

(S∞)

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 ).

(S∞)

i (·) ≥ 0.

1 up22 − uq1

1 uq22 ,

f2(u) = uq11 uq2

2 − up11 up2

2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).

1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1

1 uq22 ≤ 0.

(S∞)

i (·) ≥ 0.

1 up22 − uq1

1 uq22 ,

f2(u) = uq11 uq2

2 − up11 up2

2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).

1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1

1 uq22 ≤ 0.

(S∞)

i (·) ≥ 0.

1 up22 − uq1

1 uq22 ,

f2(u) = uq11 uq2

2 − up11 up2

2 ,p1, p2, q1, q2 ∈ [1,+∞), λ ∈ [0, 1).

1 up22 ≤ 0, f1 + λf2 = (λ− 1)uq1

1 uq22 ≤ 0.

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.

∀r ∈ [0,∞)m, Q f (r) ≤ [1 +∑

1≤i≤mri ]b,

∀r ∈ [0,∞)m, Q f (r) ≤ [1 +∑

1≤i≤mri ]b,

Existence globale pour des systèmes de réaction-diffusion ...

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