global existence for reaction-diffusion systems : a survey · jimmy lamboley,sorbonne universit...

GOAL OF THE TALKINTRODUCTION

CLASSICAL SOLUTIONSWEAK SOLUTIONS

NEW RESULTSOPEN PROBLEMS

On global solutions for reaction-diffusion systemswith bounded total mass : a survey

El-Haj LAAMRIInstitut Elie Cartan, Universite de Lorraine

TAM-TAM 9 eme editionTlemcen, 23-27 fevrier 2019

23 fevrier 2019

El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Un grand MERCI

aux Fondateurs de TAM-TAM ;

aux Membres du comite scientifique ;

aux Membres du comite d’organisation senior et junior ;

a Monsieur le Recteur de l’Universite de Tlemcen ;

Monsieur le Doyen de la faculte des sciences ;

a tous les sponsors ;

a toute personne qui a donne un coup de main, qui aencourage , ...

PDE: from theory to applications

International Workshop for the 70th birthday of Michel Pierre Nancy, France – March 25-27, 2019

Invited SpeakersNoureddine Alaa, University of Cadi Ayyad

Pascal Auscher, CNRS, LamfaKarine Beauchard, ENS Rennes

Virginie Bonnaillie-Noël , CNRSDieter Bothe, Technische Universität Darmstadt

Dorin Bucur, Université de SavoieMarc Dambrine, Université de Pau et des Pays de l'Adour

Arnaud Debussche, ENS RennesKlemens Fellner, University of Graz

Jimmy Lamboley, Sorbonne UniversitéPhilippe Laurençot, Université Paul Sabatier

Kévin Le Balc’h, ENS RennesAlain Miranville, Université de Poitiers

Arian Novruzi, University of OttawaTakashi Suzuki, University of OsakaGrégory Vial, Ecole Centrale de Lyon

Haruki Umakoshi, University of Osaka

Contact: [email protected] : //colloque-edp-2019.iecl.univ-lorraine.fr/

Organization committee:

Thibaut Deheuvels, ENS RennesHélène Jouve, El Haj Laamri, Jean Rodolphe Roche, Didier Schmitt, Université de Lorraine

Scientific committee:

Xavier Antoine, Université de LorraineGiuseppe Buttazzo, University of PisaMichel Crouzeix, Université of Rennes 1David Dos Santos Ferreira, Université de LorraineAntoine Henrot, Université de LorraineJerome Goldstein, University of MemphisJuan-Luis Vazquez, Universidad Autónoma de Madrid

Goal of the talk

Survey on global existence in time of solutions toreaction-diffusion systems (RDS) for which :• positivity is preserved ;• conservation or at least dissipation of the total mass.

This provides an a priori bound in L1, uniform in time.

QUESTION : how does this help for global existence ? ? ?

OLD AND RECENT RESULTS ; OPEN PROBLEMS

Pierre, Desvillettes ; Fellner ; Martin ; Schmitt ; Souplet ;Suzuki ; Haraux, Youkana, Barabanova, Kouachi, ... ;Abdellaoui, Attar, Bentifour, Biroud, Miri, Boukarabila,

Je ne vais parler des systemes a diffusions croisees : voir lestravaux de Bendahmane-Lepoutre-Marocco-Perthame,Boudiba-Pierre, Desvillettes-Moussa-Trescases, ...

Goal of the talk

Plan1 GOAL OF THE TALK2 INTRODUCTION

A little bit of historyA 2× 2 model systemExplicit Examples

3 CLASSICAL SOLUTIONSEasy facts on the PDE systemsThe Lp-approach

Extensions of the Lp-approachLimits of the Lp-approachWhat about initial question ?

4 WEAK SOLUTIONSLinear diffusionNonlinear degenrate diffusion

5 NEW RESULTS6 OPEN PROBLEMS

INTRODUCTION

1) A little bit of history2) A 2× 2 model system3) Explicit examples

A little bit of history

Alan Turing (1912-1954)

The Chemical Basis of Morphogenesis By A. M. Turing,University of Manchester In Phil. Trans. Royal Soc.London, 1952

Abstract : ”It is suggested that a system of chemicalsubstances, called morphogens, reacting together and diffusingthrough a tissue, is adequate to account for the mainphenomena of morphogenesis. Such a system, although it mayoriginally be quite homogeneous, may later develop a patternor structure due to an instability of the homogeneousequilibrium, which is triggered of by random disturbances...”

The Chemical Basis of Morphogenesis By A. M. Turing,University of Manchester In Phil. Trans. Royal Soc.London, 1952

Abstract : ”It is suggested that a system of chemicalsubstances, called morphogens, reacting together and diffusingthrough a tissue, is adequate to account for the mainphenomena of morphogenesis. Such a system, although it mayoriginally be quite homogeneous, may later develop a patternor structure due to an instability of the homogeneousequilibrium, which is triggered of by random disturbances...”

A 2× 2 MODEL SYSTEM

A 2× 2 model system (1)

∂tu − d1∆xu = f (u, v) in (0,T )× Ω∂tv − d2∆xv = g(u, v) in (0,T )× Ω∂u∂ν (t, x) = ∂v

∂ν (t, x) = 0 on (0,T )× ∂Ωu(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

• d1, d2 > 0 diffusion coefficients ;

• f , g : [0,+∞)2 → R regular ;

• Ω ⊂ RN , bounded open regular,

• u, v : (0,T )× Ω→ R unknown functions.

We denote QT = (0,T )× Ω and ΣT = (0,T )× ∂Ω.El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

1) A 2× 2 model system (2) : Hypotheses

∂tu − d1∆xu = f (u, v) in QT

∂tv − d2∆xv = g(u, v) in QT∂u∂ν (t, x) = ∂v

∂ν (t, x) = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

(P) Positivity preserving :∀r1, r2 ≥ 0, f (0, r2) ≥ 0, g(r1, 0) ≥ 0.(M) ’Control’ of mass : ∀r1, r2 ≥ 0,

f (r1, r2) + g(r1, r2) ≤ 0

⇒ a priori L1(Ω)-estimate, uniform in time :

∀t ≥ 0,

[u(t, x) + v(t, x)]dx ≤∫

Ω[u0(x) + v0(x)]dx .

∂ν (t, x) = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

f (r1, r2) + g(r1, r2) ≤ 0

∀t ≥ 0,

[u(t, x) + v(t, x)]dx ≤∫

Ω[u0(x) + v0(x)]dx .

∂ν (t, x) = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

f (r1, r2) + g(r1, r2) ≤ 0

∀t ≥ 0,

[u(t, x) + v(t, x)]dx ≤∫

Ω[u0(x) + v0(x)]dx .

By integrating the sum on Ω, we obtain∫Ω∂t(u(t)+v(t))dx−

(d1∆xu+d2∆xv

(f (u, v)+g(u, v))dx .

∆xu =

∫∂Ω

∂νand

∆xv =

∫∂Ω

∂ν, we have

∫Ω∂t(u(t)+v(t))dx−

∫∂Ω

(d1∂u

∂ν+d2

(f (u, v)+g(u, v))dx ≤ 0.

In our case∂u

∂ν=∂v

∂ν= 0 and f + g ≤ 0, then∫

Ω∂t(u(t) + v(t))dx ≤ 0.

(d1∆xu+d2∆xv

(f (u, v)+g(u, v))dx .

∆xu =

∫∂Ω

∂νand

∆xv =

∫∂Ω

∂ν, we have

∫∂Ω

(d1∂u

∂ν+d2

(f (u, v)+g(u, v))dx ≤ 0.

In our case∂u

∂ν=∂v

∂ν= 0 and f + g ≤ 0, then∫

Ω∂t(u(t) + v(t))dx ≤ 0.

(d1∆xu+d2∆xv

(f (u, v)+g(u, v))dx .

∆xu =

∫∂Ω

∂νand

∆xv =

∫∂Ω

∂ν, we have

∫∂Ω

(d1∂u

∂ν+d2

(f (u, v)+g(u, v))dx ≤ 0.

In our case∂u

∂ν=∂v

∂ν= 0 and f + g ≤ 0, then∫

Ω∂t(u(t) + v(t))dx ≤ 0.

∀t ∈ [0,T ] ,

[u(t, x) + v(t, x)]dx ≤∫

Ω[u0(x) + v0(x)]dx .

More generally

(M’) af (r1, r2) + bg(r1, r2) ≤ C [r1 + r2 + 1]

implies control of the total mass :

sup0≤t≤T<+∞

[u(t, x) + v(t, x)]dx < +∞.

General family of systems

Same questions for the general family of systems

1 ≤ i ≤ m∂tui − di∆xui = fi (u1, u2, · · · , um) in QT ,∂ui∂ν (t, x) = 0 on ΣT ,ui (0, ·) = u0

i (·) ≥ 0 in Ω.

di > 0, fi : [0,+∞)m → R locally Lipschitz continuous, u0i

initial data where

• (P) : positivity is preserved ; i .e the fi are quasi-positive ;

• (M) :m∑i=1

fi ≤ 0 ;. Or more generally (M’) :

∀r = (r1, r2, · · · , rm) ∈ [0,+∞)m,m∑i=1

ai fi (r) ≤ C

m∑i=1

)El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

QUESTION :Existence of global solutions in time

under (P)+(M) ? ?

or more generally under (P)+ (M’) ? ?

EXPLICIT EXAMPLES

Explicit example (1)

”Chemical morphogenetic process (“Brusselator”)

(SBruss)

∂tu − d1∆xu = −uv2 + b v in QT

∂tv − d2∆xv = uv2 − (b + 1) v + a in QT∂u∂ν (t, x) = ∂v

∂ν (t, x) = 0 on ΣT

a, b, d1, d2 > 0.

Introduced par Illya Prigogine (1917-2003), ChemistryNobel Prize in 1977.

f (0, r2) = br2 > 0, g(r1, 0) = a > 0, f + g = −r2 + a ≤ a

(SBruss)

∂ν (t, x) = 0 on ΣT

a, b, d1, d2 > 0.

f (0, r2) = br2 > 0, g(r1, 0) = a > 0, f + g = −r2 + a ≤ a

(SBruss)

∂ν (t, x) = 0 on ΣT

a, b, d1, d2 > 0.

f (0, r2) = br2 > 0, g(r1, 0) = a > 0, f + g = −r2 + a ≤ a

Explicit example (2) : Combustions models

Exothermic combustion in a gas may be modeled by a systemof the following type :

(ComMod)

∂tu − d1∆xv = −H(u, v) in QT

∂tu − d2∆xv = qH(u, v) in QT∂u∂ν (t, x) = ∂v

∂ν (t, x) = 0 on ΣT

d1, d2, q > 0.

where u is the concentration of a single reactant, v is thetemperature and H(0, r2) = 0, H(r1, 0) ≥ 0.

A typical function H is given by H(r1, r2) = rα1 er2 .

Similar equations appear for different applications.

(ComMod)

∂ν (t, x) = 0 on ΣT

d1, d2, q > 0.

(ComMod)

∂ν (t, x) = 0 on ΣT

d1, d2, q > 0.

Explicit examples 2

Lotka-Volterra Systems (Leung,...)

∀i = 1, ...,m, ∂tui − di∆xui = −eiui +∑

1≤j≤mpijuiuj ,

where ei ∈ R+, pij ∈ R such that,

∀r = (r1, r2, · · · , rm) ∈ [0,+∞[m,m∑

i ,j=1

aipij ri rj ≤ 0 for some ai > 0.

Model of diffusive calcium dynamics : H.G. Othmer∂tu1 − d1∆xu1 = λ(γ0 + γ1u4)(1− u1)− p1u4

p42+u4

∂tu2 − d2∆xu2 = −k1u2 + k ′1u3

∂tu3 − d3∆xu3 = −k ′1u3 − k2u1u3 + k1u2 + k ′2u4

∂tu4 − d4∆xu4 = k2u1u3 + k ′3u5 − k ′2u4 − k3u1u4

∂tu5 − d5∆xu5 = k3u1u4 − k ′3u5.El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Explicit examples 2

Lotka-Volterra Systems (Leung,...)

∀i = 1, ...,m, ∂tui − di∆xui = −eiui +∑

1≤j≤mpijuiuj ,

where ei ∈ R+, pij ∈ R such that,

∀r = (r1, r2, · · · , rm) ∈ [0,+∞[m,m∑

i ,j=1

aipij ri rj ≤ 0 for some ai > 0.

Model of diffusive calcium dynamics : H.G. Othmer∂tu1 − d1∆xu1 = λ(γ0 + γ1u4)(1− u1)− p1u4

p42+u4

∂tu2 − d2∆xu2 = −k1u2 + k ′1u3

∂tu3 − d3∆xu3 = −k ′1u3 − k2u1u3 + k1u2 + k ′2u4

∂tu4 − d4∆xu4 = k2u1u3 + k ′3u5 − k ′2u4 − k3u1u4

∂tu5 − d5∆xu5 = k3u1u4 − k ′3u5.El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Explicit examples 3 : Reversible chemical reactions

Model of chemical reactions 1 (mass action law)

U1 + U3 U2 + U4

(Sq,4×4)

∂tu1 − d1∆xu1 = −u1u3 + u2u4 = f1(u1, u2, u3, u4)∂tu2 − d2∆xu2 = u1u3 − u2u4 = f2(u1, u2, u3, u4)∂tu3 − d3∆xu3 = −u1u3 + u2u4 = f3(u1, u2, u3, u4)∂tu4 − d4∆xu4 = u1u3 − u2u4 = f4(u1, u2, u3, u4)

Here f1 + f2 + f3 + f4 = 0.

(Sq,4×4)

1 ≤ i ≤ 4∂tui − di∆xui = (−1)i (u1u3 − u2u4) in QT∂ui∂ν

(t, x) = 0 on ΣT

ui (0, x) = u0i (x) ≥ 0 in Ω.

Explicit examples 3 : Reversible chemical reactions

Model of chemical reactions 2 (mass action law)

αU + βV γW

∂tu − d1∆xu = α(wγ − uαvβ) = f1(u, v ,w)∂tv − d2∆xv = β(wγ − uαvβ) = f2(u, v ,w)∂tw − d3∆xw = γ(−wγ + uαvβ) = f3(u, v ,w).

α, β, γ ≥ 1.

En chimie, α, β, γ sont des entiers naturels.

Here βγf1 + αγf2 + 2αβf3 = 0.

Easy facts on the PDE systemsThe Lp -approach

Extensions of the Lp -approachLimits of the Lp -approachWhat about initial question ?

CLASSICAL SOLUTIONS

Easy facts on the PDE systems

Lp-approach

Limits of the Lp-approach

Some new results

Open problems

Easy facts on the PDE systems (1)

∂tv − d2∆xv = g(u, v) in QT∂u∂ν = ∂v

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

For u0, v0 ∈ L∞(Ω), local existence and uniqueness ofclassical solution (u, v) to (MS) are known.

More precisely, there exists T ∗ > 0 and a unique classicalsolution (u, v) of (MS) on [0,T ∗)× Ω.

Moreover u, v ≥ 0 thanks to (P).

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

The maximal time of existence T ∗ is characterized by

supt∈(0,T∗)

(‖u(t, .)‖L∞(Ω)+‖v(t, .)‖L∞(Ω)

)< +∞ ⇒ T ∗ = +∞.

A priori L∞-estimates, uniform in time, implies globalexistence !

But this type of estimates is far of being obvious for oursystem except in the case where diffusion coefficientsare egal.

supt∈(0,T∗)

(‖u(t, .)‖L∞(Ω)+‖v(t, .)‖L∞(Ω)

)< +∞ ⇒ T ∗ = +∞.

supt∈(0,T∗)

(‖u(t, .)‖L∞(Ω)+‖v(t, .)‖L∞(Ω)

)< +∞ ⇒ T ∗ = +∞.

Case : d1 = d2 = d

∂t [u + v ]− d∆x(u + v) = f (u, v) + g(u, v) ≤ 0.

By maximum principle for the heat equation :

∀t ∈ (0,T ∗), ‖u(t, .) + v(t, .)‖L∞(Ω) ≤ ‖u0 + v0‖L∞(Ω).

and, since 0 ≤ u, v , this implies L∞-bounds on u(t, .), v(t, .) andtherefore

T ∗ = +∞ !!

Later on, we will assume d1 6= d2.

Lp-approach (1)

Recall the Brusselator introduced by Illya Prigogine (β ≥ 1)

(Bruss)

∂tu − d1∆xu = −uvβ ≤ 0 in QT

∂tv − d2∆xv = uvβ in QT∂u∂ν = ∂v

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

Then, by maximum principle :∀t ∈ (0,T ∗), ‖u(t, .)‖L∞(Ω) ≤ ‖u0‖L∞(Ω).

But what about v ?

∀t ∈ (0,T ∗), ‖v(t, .)‖L∞(Ω) ≤ C and therefore T ∗ = +∞.

Alikakos (β <N + 2

N), Mazuda, Hollis-Martin-Pierre, ...

Lp-approach (1)

(Bruss)

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

But what about v ?

Alikakos (β <N + 2

Lp-approach (1)

(Bruss)

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

But what about v ?

Alikakos (β <N + 2

Lp-approach (1)

(Bruss)

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

But what about v ?

Alikakos (β <N + 2

Lp-approach (1)

(Bruss)

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0.

But what about v ?

Alikakos (β <N + 2

Lp-approach applied to the Brusselator (1)

Ingredient : The Lp-estimate by duality introduced by MichelPierre (1985)

Theorem (Pierre’s duality lemma)

Let u, v ≥ 0 such that

∂tu − d1∆u + ∂tv − d2∆v ≤ 0.

Then there exists C = C (p,T ,Ω) such that

∀1 < p < +∞, ‖v‖Lp(QT ) ≤ C (1 + ‖u‖Lp(QT )).

From the fundamental Lemma, for all 1 < p <∞‖v‖Lp(QT∗ ) ≤ C‖u‖Lp(QT∗ ) ≤ C‖u0‖L∞(Ω).

Lp-approach applied to the Brusselator (1)

Ingredient : The Lp-estimate by duality introduced by MichelPierre (1985)

Theorem (Pierre’s duality lemma)

Let u, v ≥ 0 such that

∂tu − d1∆u + ∂tv − d2∆v ≤ 0.

Then there exists C = C (p,T ,Ω) such that

∀1 < p < +∞, ‖v‖Lp(QT ) ≤ C (1 + ‖u‖Lp(QT )).

From the fundamental Lemma, for all 1 < p <∞‖v‖Lp(QT∗ ) ≤ C‖u‖Lp(QT∗ ) ≤ C‖u0‖L∞(Ω).

The Lp-approach applied to the Brusselator (2)

For all 1 < q <∞

‖uvβ‖Lq(QT∗ ) ≤ C .

Choosing q large enough ( q >N + 1

2), this implies

‖v‖L∞(QT∗ ) ≤ C .

Ceci est du au theoreme de regularite maximale.

Then T ∗ =∞.

For all 1 < q <∞

2), this implies

‖v‖L∞(QT∗ ) ≤ C .

Then T ∗ =∞.

For all 1 < q <∞

2), this implies

‖v‖L∞(QT∗ ) ≤ C .

Then T ∗ =∞.

Extension 1

This approach extends to m ×m-systems

1 ≤ i ≤ m;∂tui − di∆xui = fi (u1, u2, · · · , um) in QT∂ui∂ν (t, x) = 0 on ΣT

ui (0, ·) = u0i (·) ≥ 0 in Ω

where the fi have polynomial growth and

f1 ≤ 0f1 + f2 ≤ 0f1 + f2 + f3 ≤ 0...f1 + f2 + · · ·+ fm ≤ 0.

More generally

There exists an invertible matrix Q ∈Mm((0,+∞)) and

b2...bm

∈Mm,1((0,+∞)) such that for all

r = (r1, r2, · · · , rm) ∈ [0,+∞)m, we have

Qf(r) ≤

where f :=

f1f2...fm

Extention 2

RecallαU + βV γW

(Sαβγ)

∂tu − d1∆u = α(wγ − uαvβ) = f1(u, v ,w) in QT ,∂tv − d2∆v = β(wγ − uαvβ) = f2(u, v ,w) in QT ,∂tw − d3∆w = γ(−wγ + uαvβ) = f3(u, v ,w) in QT ,∂u

∂ν=∂v

∂ν=∂w

∂ν= 0 on ΣT ,

u(0, .) = u0 ≥ 0 ; v(0, .) = v0 ≥ 0 ; w(0, .) = v0 ≥ 0 in ∈ Ω.

α, β, γ ≥ 1

Extention 2

Previously, the following cases have been studied by severalauthors :

Case α = β = γ = 1 :• Rothe for dimensions N ≤ 5 (1984).• Pierre, Morgan, Feng, Kouachi ..., for all dimensions N

Case γ = 1 and arbitrary α, β ≥ 1 :EHL, Martin -Pierre

Case γ > α + β :EHL

E.-H. Laamri : Global existence of classical solutions for aclass of reaction-diffusion systems, Acta Appl. Math. 115(2011), no. 2, 153-165.

The case 2 ≤ γ ≤ α + β OPEN

Extention 2

Limits of the Lp-approach

Limits of the Lp-approach (1)

The Lp-approach does not apply to

(Sexp)

∂tu − d1∆xu = −uevβ

∂tv − d2∆xv = uevβ

in QT∂u∂ν = ∂v

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

(Spenible)

∂tu − d1∆xu = u3v2 − u2v3 in QT

∂tv − d2∆xv = −u3v2 + u2v3 in QT∂u∂ν = ∂v

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

The Lp-approach does not apply to

(Sexp)

∂tu − d1∆xu = −uevβ

∂tv − d2∆xv = uevβ

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

(Spenible)

∂tu − d1∆xu = u3v2 − u2v3 in QT

∂tv − d2∆xv = −u3v2 + u2v3 in QT∂u∂ν = ∂v

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

The Lp-approach does not apply to nonlinear diffusion.

Let consider the following system

(NLDSS)

∂tu − d1∆xu

β = −g(u, v) in QT

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω

If 0 ≤ g(u, v) ≤ ψ(u) (1 + vγ), (NLDSS) has a globalclassical solution.

What’s happen in the others cases ?

The Lp-approach does not apply to nonlinear diffusion.

Let consider the following system

(NLDSS)

∂tu − d1∆xu

β = −g(u, v) in QT

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω

If 0 ≤ g(u, v) ≤ ψ(u) (1 + vγ), (NLDSS) has a globalclassical solution.

What’s happen in the others cases ?

What about initial question ?

QUESTION : Existence of classical solutions in timeunder (P)+(M) ? ?

Theorem (Pierre-Schmitt ’97)

One can find polynomials nonlinearites f , g satisfying (P), (M)(f + g ≤ 0) and also

∃λ ∈ [0, 1[ such that f + λg ≤ 0

for which there exists T ∗ < +∞ with

limt→T∗

‖u(t, .)‖L∞(Ω) = limt→T∗

‖v(t, .)‖L∞(Ω) = +∞.

QUESTION : Existence of classical solutions in timeunder (P)+(M) ? ?

Theorem (Pierre-Schmitt ’97)

One can find polynomials nonlinearites f , g satisfying (P), (M)(f + g ≤ 0) and also

∃λ ∈ [0, 1[ such that f + λg ≤ 0

for which there exists T ∗ < +∞ with

limt→T∗

‖u(t, .)‖L∞(Ω) = limt→T∗

‖v(t, .)‖L∞(Ω) = +∞.

Blow-up may appear even in space dimension N = 1 withhigh degree polynomial nonlinearites.

Blow-up may appear with any superquadratic growth 2 + ε forf and g (with high dimension N).

M. Pierre ; D. Schmitt : Blow-up in reaction-diffusionsystems with dissipation of mass, SIAM J. Math. Ana. 28no 2 (1997), 259-269.

M. Pierre ; D. Schmitt : Blow-up in reaction-diffusionsystems with dissipation of mass, SIAM Rev. 42 no 2(2000), 93-106 (electronic).

Blow-up may appear even in space dimension N = 1 withhigh degree polynomial nonlinearites.

Blow-up may appear with any superquadratic growth 2 + ε forf and g (with high dimension N).

M. Pierre ; D. Schmitt : Blow-up in reaction-diffusionsystems with dissipation of mass, SIAM J. Math. Ana. 28no 2 (1997), 259-269.

M. Pierre ; D. Schmitt : Blow-up in reaction-diffusionsystems with dissipation of mass, SIAM Rev. 42 no 2(2000), 93-106 (electronic).

Linear diffusionNonlinear degenrate diffusion

WEAK SOLUTIONS

LINEAR DIFFUSION

NONLINEAR DIFFUSION

NEW RESULTS

OPEN PROBLEMS

LINEAR DIFFUSION

L1-Theorem and Applications

L2-Theorem and Applications

Open problems

Linear diffusion

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

NOW we will look for :WEAK SOLUTIONS (which may go out of L∞(Ω) at some

times)... but the nonlinearities f (u, v), g(u, v) are in L1(QT )for all T > 0.This is sometimes called INCOMPLETE BLOW UP !

...like u(t, x) =1

t2 + |x |2which is weak solution of

∂tu −∆u = c(t, x) u2 with c(t, x) bounded (N ≥ 4).El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Linear diffusion

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

NOW we will look for :WEAK SOLUTIONS (which may go out of L∞(Ω) at some

times)... but the nonlinearities f (u, v), g(u, v) are in L1(QT )for all T > 0.This is sometimes called INCOMPLETE BLOW UP !

...like u(t, x) =1

t2 + |x |2which is weak solution of

∂tu −∆u = c(t, x) u2 with c(t, x) bounded (N ≥ 4).El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Notion of solution

Recall the model system

∂ν = 0 on ΣTu(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω.

Weak solution means : f (u, v), g(u, v) are in L1(QT ) for allT > 0 and (MS) is understood in the sense of distributions or,equivalently here, solution in the sense of correspondingsemigroups

u(t, x) = Sd1(t)u0 +

0Sd1(t − s)f (u, v) ds

v(t, x) = Sd2(t)v0 +

0Sd2(t − s)g(u, v) ds

where Sdi (.) is the semigroup generated in L1(Ω) by −di∆ withhomogeneous Neumann boundary condition.

L1-THEROEM FOR LINEAR DIFFUSIONAND APPLICATIONS

L1-theorem for linear diffusion

Theorem (Pierre, 2003, L1-Theorem )

Assume (P)+ (M) hold together with the a priori estimate

∀T > 0,

[|f (u, v)|+ |g(u, v)|

]dt dx ≤ C (T ) < +∞.

Then, (MS) has a global weak solution for all(u0, v0) ∈ (L1(Ω)+)2.

Proof involves truncations and L1-type estimates like∫[0≤u≤k]

d1|∇u|2 ≤ k

(∫Ω|f (u, v)|+

∫Ωu0

Theorem (Pierre, 2003, L1-Theorem )

Assume (P)+ (M) hold together with the a priori estimate

∀T > 0,

[|f (u, v)|+ |g(u, v)|

]dt dx ≤ C (T ) < +∞.

Then, (MS) has a global weak solution for all(u0, v0) ∈ (L1(Ω)+)2.

Proof involves truncations and L1-type estimates like∫[0≤u≤k]

d1|∇u|2 ≤ k

(∫Ω|f (u, v)|+

∫Ωu0

Estimate∫QT|f |+ |g | ≤ C

Proposition

Assume (P) and

(M) f + g ≤ 0 AND (Mλ) f + λg ≤ 0 for some λ < 1.

Then, if (u, v) is solution of (MS) on (0,T ),∫QT

[|f (u, v))|+ |g(u, v)|] dt dx ≤ M = M(data) < +∞.

That is to say : the nonlinearities f (u, v), g(u, v) are a prioribounded in L1(QT ) for all T > 0.

Application of L1-theorem for linear diffusion

Recall the System with exponential growth

(Sexp)

∂tu − d1∆xu = −uαevβ

∂tv − d2∆xv = uαevβ

∂ν = 0 on ΣT

u(0, ·) = u0(·) ≥ 0, v(0, ·) = v0(·) ≥ 0 in Ω

has global classical solution for any β ≤ 1 and for nonnegativeinitial data (u0, v0) ∈ L∞(Ω)× L∞(Ω) and

‖u0‖L∞(Ω) ≤8d1d2

αN(d1 − d2)2.

Haraux-Youkana for β < 1 and Barabanova for β = 1.

Thanks to L1-theorem, System (Sexp) has global weaksolution for any β > 0 and for any nonnegative initial data(u0, v0) ∈ L1(Ω)× L1(Ω).Easy L1(QT )-estimate of the nonlinearity :∫

Ωu(T , x) dx +

uαevβdtdx =

∫Ωu0(x) dx .

Recall that the equation

(Eexp)

∂tw −∆xw = ew

2in QT

∂ν= 0 on ΣT

w(0, ·) = w0(·) ≥ 0 in Ω

does not have even local solution in general when w0 isonly integrable on Ω.

Ωu(T , x) dx +

uαevβdtdx =

∫Ωu0(x) dx .

(Eexp)

∂tw −∆xw = ew

2in QT

∂ν= 0 on ΣT

w(0, ·) = w0(·) ≥ 0 in Ω

Ωu(T , x) dx +

uαevβdtdx =

∫Ωu0(x) dx .

(Eexp)

∂tw −∆xw = ew

2in QT

∂ν= 0 on ΣT

w(0, ·) = w0(·) ≥ 0 in Ω

More generally

L1-theorem holds for systems with m ≥ 2 equations

(RDSLD)

1 ≤ i ≤ m

∂tui − di∆xui = fi (u1, · · · , um) QT ,∂ui∂ν = 0 ΣT

ui (0, x) = u0,i (x) ≥ 0 x ∈ Ω,

where the nonlinearites fi satisfy (P), (M’) and thefollowing L1− a priori estimate

(*) sup1≤i≤m

‖fi (u)‖L1(QT ) ≤ C (T ) for all T > 0.

Remark : the following L1− a priori estimate (*) is satified ifthere are m independent inequalites between the fi ’s.

L1-theorem and Reversible chemical reactions

U1 + U3 U2 + U4

(Sq,4×4)

1 ≤ i ≤ 4∂tui − di∆xui = (−1)i (u1u3 − u2u4) in QT∂ui∂ν

(t, x) = 0 on ΣT

ui (0, x) = u0i (x) ≥ 0 in Ω.

αU + βV γW

(Sαβγ)

∂tu − d1∆xu = α(wγ − uαvβ) = f1(u, v ,w)∂tv − d2∆xv = β(wγ − uαvβ) = f2(u, v ,w)∂tw − d3∆xw = γ(−wγ + uαvβ) = f3(u, v ,w).

α, β, γ ≥ 1.El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

L2-THEROEM FOR LINEAR DIFFUSIONAND APPLICATIONS

(RDSLD)

1 ≤ i ≤ m∂tui − di∆xui = fi (u1, · · · , um) in (QT ,∂ui∂ν (t, x) = 0 on ΣT

ui (0, x) = u0,i (x) in Ω,

Theorem (Michel Pierre, L2-theorem)

Assume (P)+(M) and ∀1 ≤ i ≤ m, ∀r ∈ [0,+∞)m

|fi (r1, · · · , rm)| ≤ C (1 +∑

1≤j≤mr2j ).

Then , System (RDSLD) has a global weak solution for anynonnegative initial data u0 ∈ (L2(Ω)+)m.

Proof of L2-theorem for linear diffusion

Lemma (Pierre-Schmitt, 97’, L2-Estimate)

Assume f = (f1, · · · , fm) satisfies (P) and (M). Then, thefollowing a priori estimate holds for solutions of (RDSLD) :

∀1 ≤ i ≤ m, ∀T > 0, ‖ui‖L2(QT ) ≤ C = C

T , di ,∑j

‖u0,j‖L2(Ω)

Proof : By L2-theorem + at most quadratic growth of fi , wehave the a priori estimate for the solutions of System(RDSLD)

∀1 ≤ i ≤ m,

|fi (u)|dtdx ≤ C (T ).

Then , we apply L1-theorem.

Proof of L2-theorem for linear diffusion

Lemma (Pierre-Schmitt, 97’, L2-Estimate)

Assume f = (f1, · · · , fm) satisfies (P) and (M). Then, thefollowing a priori estimate holds for solutions of (RDSLD) :

∀1 ≤ i ≤ m, ∀T > 0, ‖ui‖L2(QT ) ≤ C = C

T , di ,∑j

‖u0,j‖L2(Ω)

Proof : By L2-theorem + at most quadratic growth of fi , wehave the a priori estimate for the solutions of System(RDSLD)

∀1 ≤ i ≤ m,

|fi (u)|dtdx ≤ C (T ).

Then , we apply L1-theorem.

Applications of L2-theorem to quadratic systems with lineardiffusion

Applies to the reversible chemical reaction :U1 + U3 U2 + U4.

for 1 ≤ i ≤ 4, ∂tui − di∆xui = (−1)i (u1u3 − u2u4).

Desvillettes-Fellner-Pierre-Vovelle (2007).Initial data in (L2(Ω)+)4.

Applies to Lotka-Volterra Systems :

∀i = 1, ...,m, ∂tui − di∆xui = −eiui +∑

1≤j≤mpijuiuj ,

Pierre

and many others ... For more details, see Pierre’s survey,Milan J. Math. 78, no. 2 (2010), 417-455.

∀i = 1, ...,m, ∂tui − di∆xui = −eiui +∑

1≤j≤mpijuiuj ,

Pierre

∀i = 1, ...,m, ∂tui − di∆xui = −eiui +∑

1≤j≤mpijuiuj ,

Pierre

∀i = 1, ...,m, ∂tui − di∆xui = −eiui +∑

1≤j≤mpijuiuj ,

Pierre

NONLINEAR DEGENERATE DIFFUSION

Extension of L1-Theorem

Extension of L2-Theorem and Applications

Open problems

New result : Nonlinear Degenerate Diffusion (1)

• Question : What about systems with NonlinearDegenerate Diffusion of porous media type ∆x(uβii ) with

βi ∈] (N−2)+

N ,+∞) ?

(RDSNLD)

1 ≤ i ≤ m

∂tui − di∆x(uβii ) = fi (u1, · · · , um) QT ,ui (t, x) = 0 ΣT

ui (0, x) = u0,i (x) ≥ 0 x ∈ Ω,

where di ∈ (0,+∞).•

E.-H. Laamri, M. Pierre : Global existence forreaction-diffusion systems with nonlinear diffusion and controlof mass, Ann. Inst. H. Poincare-Anal. Non Lineaire 34 (2017),571-591.

Nonlinear degenerate diffusion (2)

Theorem (Pierre, E-H.L, 2016)

Assume f = (f1, · · · , fm) satisfies (P), (M’) and assume there isan L1− a priori estimate on the nonlinearities

sup1≤i≤m

‖fi (u)‖L1(QT ) ≤ C (T ) for allT > 0.

Then, for βi < 2, there exists a global weak solution to(RDSNLD) for all u0 = (u0,1, · · · , u0,m) ∈ (L1(Ω)+)m.

∫[0≤ui≤k]

di |∇ui |2 ≤ k2−βi(∫

Ω|fi (u)|+

∫Ωu0,i

OPEN PROBLEM : What happens when βi ≥ 2 ?El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Extention of L2-estimates

Lβi+1-estimates

(RDSNLD)

1 ≤ i ≤ m∂tui −∆x(ui )

βi = fi (u1, · · · , um) in QT ,ui (t, x) = 0 on ΣT ,ui (0, x) = u0,i (x) ≥ 0 in Ω.

Proposition (Pierre, E-H.L, 2016)

Assume f = (f1, · · · , fm) satisfies (P) and (M’). Then , thefollowing a priori estimate holds for solutions of (RDSNLD) :∀1 ≤ i ≤ m, ∀T > 0

‖ui‖Lβi+1(QT ) ≤ C = C

T ,∑j

‖u0,j‖L2(Ω)

Application of Lβi+1-estimates

Corollary (M. Pierre, E-H.L, 2016)

Assume f = (f1, · · · , fm) satisfies (P), (M) and

|fi (r1, · · · , rm)| ≤ C

(rj)βj+1−ε + 1

Then, there exists a global weak solution to (SRNLD) for allu0 = (u0,1, · · · , u0,m) ∈ (L2(Ω)+)m.

Application of Lβi -estimate to (favorite) quadratic system

U1 + U3 U2 + U4

(QRDSNLD)

1 ≤ i ≤ 4,∂tui − di∆x(ui )

βi = (−1)i (u1u3 − u2u4) in QT ,ui (t, x) = 0 in ΣT ,ui (0, x) = u0,i (x) ≥ 0 in Ω.

Theorem (M. Pierre, E-H.L, 2016)

Assume that u0 = (u0,1, u0,2, u0,3, u0,4) ∈ (L2(Ω)+)4. Then, thesystem (QRDSNLD) has a global weak solution for allβi ∈ [1,+∞[.

- if min1≤i≤4

βi > 1, we apply the corollary ;

- iff min1≤i≤4

βi = 1, entropy inequality implies uniform integrablity.El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

Application of Lβi -estimate to more general reversiblereaction

More generally, let us consider the following reversible reaction

p1U1 + p2U2 + · · ·+ pmUm q1U1 + q2U2 + · · ·+ qmUm

∀1 ≤ i ≤ m, ∂tui −∆x(ui )βi = (pi − qi )

uqjj −

Global existence of weak solution holds if [E.H.L ; Michel

Pierre, 2016]∑i

piβi + 1

≤ 1 and∑i

qiβi + 1

≤ 1.

Question : What about existence in general ?

∀1 ≤ i ≤ m, ∂tui −∆x(ui )βi = (pi − qi )

uqjj −

Pierre, 2016]∑i

piβi + 1

≤ 1 and∑i

qiβi + 1

≤ 1.

∀1 ≤ i ≤ m, ∂tui −∆x(ui )βi = (pi − qi )

uqjj −

Pierre, 2016]∑i

piβi + 1

≤ 1 and∑i

qiβi + 1

≤ 1.

NEW RESULTS 2018

Stationnary systems EHL-Pierre

Pierre-Umakoshi

Philippe Souplet

Fellner et all

Stationnary Systems by EHL-Pierre (1)

Approximating by stationnary reaction-diffusion systems.• Recall that the implicit time-discetrisation of

∂tui − di∆uβii = fi (u1, u2, · · · , um), i = 1, · · · ,m.

is given , for h : tn+1 − tn, by

ui (tn+1)− hdi∆(ui (tn+1))βi = h[fi (u(tn+1)] + ui (tn).

• This yields to the question of existence for stationnaryreaction-diffusion systems of type :

i = 1, · · · ,mui − δi∆uβii = Fi (u1, u2, · · · , um) + gi

with nonlinearities Fi satisfying (P), (M) or (M’) and variousboundary conditions.

Stationnary Systems by EHL-Pierre (2)

• Or more generally

(ERDS)

i = 1, · · · ,m ,gi ∈ L1(Ω)+ ,ui + Aiui = Fi (u1, u2, · · · , um) + gi ,

where the Aiui are good m-accretive nonlinear diffusion operators.

RESULTS :

• 1) Existence of weak solution (Fi (u) ∈ L1(Ω) and the Fi satisfym independant inequalities.This includes Aiui = −∆uβii with βi ∈ [1,+∞[ and classical

boundary conditions.

Stationnary Systems by EHL-Michel PIERRE (3)

• 2) For linear diffusions and “chemical” nonlinearities

Fi (u) = λi[∏

uqjj −

], λi (pi − qi ) > 0.

Existence of weak solutions for gi log gi ∈ L1(Ω) and m ≤ 5 .

Questions :- Extension of (2) to any m, gi ∈ L1, nonlinear diffusions.- Exploit the estimates on the stionnary case to go back to globalexistence for the evolution case. -

LAAMRI, El Haj ; PIERRE, Michel : Stationaryreaction–diffusion systems in L1. Math. Models Methods Appl.Sci. 28 (2018), no. 11, 2161–2190.

Pierre-Haruki

Recall

(RDSNLD)

1 ≤ i ≤ m

∂tui − di∆x(uβii ) = fi (u1, · · · , um) QT ,ui (t, x) = 0 ΣT ,ui (0, x) = u0,i (x) ≥ 0 x ∈ Ω,

where di ∈ (0,+∞).

EHL, Michel PIERRE (2016), global existence for any m and

βi ∈] (N−2)+

N , 2[Michel PIERRE-Haruki UMAKOSHI (2018) : global existence form = 2 and βi ∈]0, 2].

New results 2018-2019 : RDS with quadratic growth

Rappel : U1 + U3 U2 + U4

(Sq,4×4)

∂tu1 − d1∆xu1 = u2u4 − u1u3 ]0,T [×Ω,∂tu2 − d2∆xu2 = −u2u4 + u1u3 ]0,T [×Ω,∂tu3 − d3∆xu3 = u2u4 − u1u3 ]0,T [×Ω,∂tu4 − d4∆xu4 = −u2u4 + u1u3 ]0,T [×Ω,

∂ui∂ν = 0 (0,T )× ∂Ω,

ui (0, x) = u0,i (x) x ∈ Ω.

• Global existence of weak solution for all N and

initial data in L2 by Desvillettes-Fellner-Pierre-Vovelle(2007).initial data in L1 by Pierre-Rolland (2017).

• Global existence of classical solution for N = 1, 2 and boundedinitial data by Pruss ; Goudon-Vasseur ; Caniso-Desvillettes-Fellner .• Open for classical solutions when N ≥ 3.El-Haj LAAMRI Institut Elie Cartan, Universite de Lorraine On global solutions for reaction-diffusion systems with bounded total mass : a survey

New result 2018 : Philippe Souplet (1)

(RDSLD)

1 ≤ i ≤ m∂tui −∆x(ui ) = fi (u1, · · · , um) (0,T )× Ω,ui (t, x) = 0 (0,T )× ∂Ωui (0, x) = u0,i (x) ≥ 0 x ∈ Ω,

Assume that• initial data u0 = (u0,1, u0,2 · · · , u0,m) ∈ (L∞(Ω)+)m ;• the nonlinearities fi satisfy (P), (M), and for allr = (r1, r2 · · · , rm) ∈ [0,+∞[m

• Quadratic growth |fi (r)| ≤ C [1 +∑j

(rj)2] ;

• Entropy dissipation∑i

fi (r) ln ri ≤ 0.

New result 2018 : Philippe Souplet (2)

Theorem (Philippe Souplet (2018))

Assume that the fi satisfy Quasi-positivity, Mass dissipation,Quadratic growth and Entropy dissipation. Then, System(RDSLD) has a global classical solution for all N.

Philippe Souplet : Global existence for reaction-diffusionsystems with dissipation of mass and quadratic growth, J.Evol. Equ. (2018).

SOME OPEN PROBLEMS (1)

1) Going back to the initial system with nonlinear diffusion

(RDSNLD)

1 ≤ i ≤ m

∂tui −∆x(uβii ) = fi (u1, · · · , um) (0,T )× Ω,ui (t, x) = 0 (0,T )× ∂Ωui (0, x) = u0,i (x) ≥ 0 x ∈ Ω,

What’s happen if max1≤i≤m

βi ≥ 2 ?

2) If we have ∆pi (ui ) = div(|∇ui |pi−2|∇ui |) instead ∆x(ui )βi ?

3) What’s happen if the diffusion are driven by operators offractional type ?

4) Once we have proved global existence of weak solutions for asystem, it remains interesting to decide whether it is unformlybounded (and therefore classical) or not. Let us for instance thecombustion model

∂tu − d1∆u = −uα ev2

∂tv − d2∆v = uα ev2.

Note that, as proved by Rebiai-Benachour, L∞-bounds hold for theslightly better system

f (u, v) = −uα(1 + v2)ev2, g(u, v) = uα ev

5) How far is it possible to extend the results recalled in this talkto situations where the nonlinearities depend also on the gradientof the solutions, like they do in several models ? A 2× 2 modelwould be of the form

∂tu − d1∆u = f (u, v ,∇u,∇v)∂tv − d2∆v = g(u, v ,∇u,∇v),

together with conditions of the kind f + g ≤ 0. All questions aboutglobal existence of classical, or of weak solutions, are of interest.

6) How do the known techniques extend to cross-diffusions, namely∂tu − d1∆u −∇ · (a1(u, v)∇u + a2(u, v)∇v) = f (u, v) on QT

∂tv − d2∆v −∇ · (b1(u, v)∇u + b2(u, v)∇v) = g(u, v) on QT ?

More and more pertinent models require these cross-diffusions.Conditions are required to preserve positivity. Next, globalexistence remains a natural question.

7) Instead of having an L1-structure of type (M), a f + b g ≤ 0with a, b > 0, there are systems for which a more generalLyapunov structure holds like

h′1(u)f (u, v) + h′2(v)g(u, v) ≤ 0,

where h1, h2 : [0,+∞)→ [0,+∞) satisfy limr→∞

hi (r) = +∞. Global

existence would still hold for the associated O.D.E. But, whatabout the P.D.E. system, even for convex hi ? Some results may bededuced directly from the remark that, due to the convexity of thehi , h(u) = (h1(u1), .., hm(um)) is a subsolution of a systemsatisfying (P)+(M).

Analogie des EDP de la physique, POINCARE (1)

Quand on envisage les divers problemes de Calcul Integral qui seposent naturellement lorsqu’on veut approfondir les parties les plusdifferentes de la Physique, il est impossible de n’etre pas frappe desanalogies que tous ces problemes presentent entre eux. Qu’ils’agisse de l’electricite statique ou dynamique, de la propagation dela chaleur, de l’optique, de l’elasticite, de l’hydrodynamiquetoujours conduit a des EDP de meme famille et les conditions auxlimites, quoique differentes, ne sont pas pourtant sans offrirquelques ressemblances, [...] un certain air de famille.

Analogie des EDP de la physique, POINCARE (2)

On doit donc s’attendre a leur trouver un tres grand nombre deproprietes communes. Malheureusement la premiere des proprietescommunes a tous ces problemes, c’est leur extreme difficulte. Nonseulement on ne peut le plus souvent les resoudre completement,mais ce n’est qu’au prix des plus grandes difficultes qu’on peut endemontrer rigoureusement la possibilite.

Bibliography

E.-H. Laamri : Global existence of classical solutions for a classof reaction-diffusion systems, Acta Appl. Math. 115 (2011),no. 2, 153-165.

E.-H. Laamri, M. Pierre : Global existence forreaction-diffusion systems with nonlinear diffusion and controlof mass, Ann. Inst. H. Poincare-Anal. Non Lineaire 34 (2017),571-591.

K. Fellner, E.-H. Laamri : Exponential decay towardsequilibrium and global classical solutions for nonlinearreaction-diffusion systems, J. Evol. Equ. 16 (2016), no. 3,681-704.

M. Pierre, Global Existence in Reaction-Diffusion Systems withDissipation of Mass : a Survey, Milan J. Math. 78, no. 2(2010), 417-455.

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