Construction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

Translated title of the contribution: Construction and stability of blowup solutions for a non-variational semilinear parabolic system

Tej-eddine Ghoul, Van Tien Nguyen, Hatem Zaag

Research output: Contribution to journalArticle

Abstract

We consider the following parabolic system whose nonlinearity has no gradient structure: {∂tu=Δu+|v|p−1v,∂tv=μΔv+|u|q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics: {u(x,t)∼Γ[(T−t)[Formula presented],v(x,t)∼γ[(T−t)[Formula presented], with b=b(p,q,μ)>0 and (Γ,γ)=(Γ(p,q),γ(p,q)). The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ=1; and the fact that the case μ≠1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

Original languageFrench
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
DOIs
StateAccepted/In press - Jan 1 2018

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Semilinear Parabolic Systems
Blow-up of Solutions
Blow-up
Index Theory
Parabolic Systems
Nonlinearity
Gradient
Perturbation
Symmetry
Operator

Keywords

  • Blowup profile
  • Blowup solution
  • Semilinear parabolic system
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics

Cite this

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title = "Construction et stabilit{\'e} de solutions explosives pour un syst{\`e}me parabolique s{\'e}milin{\'e}aire non-variationel",
abstract = "We consider the following parabolic system whose nonlinearity has no gradient structure: {∂tu=Δu+|v|p−1v,∂tv=μΔv+|u|q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics: {u(x,t)∼Γ[(T−t)[Formula presented],v(x,t)∼γ[(T−t)[Formula presented], with b=b(p,q,μ)>0 and (Γ,γ)=(Γ(p,q),γ(p,q)). The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ=1; and the fact that the case μ≠1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.",
keywords = "Blowup profile, Blowup solution, Semilinear parabolic system, Stability",
author = "Tej-eddine Ghoul and Nguyen, {Van Tien} and Hatem Zaag",
year = "2018",
month = "1",
day = "1",
doi = "10.1016/j.anihpc.2018.01.003",
language = "French",
journal = "Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis",
issn = "0294-1449",
publisher = "Elsevier Masson SAS",

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TY - JOUR

T1 - Construction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

AU - Ghoul, Tej-eddine

AU - Nguyen, Van Tien

AU - Zaag, Hatem

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider the following parabolic system whose nonlinearity has no gradient structure: {∂tu=Δu+|v|p−1v,∂tv=μΔv+|u|q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics: {u(x,t)∼Γ[(T−t)[Formula presented],v(x,t)∼γ[(T−t)[Formula presented], with b=b(p,q,μ)>0 and (Γ,γ)=(Γ(p,q),γ(p,q)). The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ=1; and the fact that the case μ≠1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

AB - We consider the following parabolic system whose nonlinearity has no gradient structure: {∂tu=Δu+|v|p−1v,∂tv=μΔv+|u|q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics: {u(x,t)∼Γ[(T−t)[Formula presented],v(x,t)∼γ[(T−t)[Formula presented], with b=b(p,q,μ)>0 and (Γ,γ)=(Γ(p,q),γ(p,q)). The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ=1; and the fact that the case μ≠1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

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