### Abstract

We consider the following parabolic system whose nonlinearity has no gradient structure: {∂_{t}u=Δu+|v|^{p−1}v,∂_{t}v=μΔv+|u|^{q−1}u,u(⋅,0)=u_{0},v(⋅,0)=v_{0}, in the whole space R^{N}, 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(u_{0},v_{0}) 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 language | French |
---|---|

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

### Fingerprint

### Keywords

- Blowup profile
- Blowup solution
- Semilinear parabolic system
- Stability

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics

### Cite this

**Construction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel.** / Ghoul, Tej-eddine; Nguyen, Van Tien; Zaag, Hatem.

Research output: Contribution to journal › Article

}

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.

KW - Blowup profile

KW - Blowup solution

KW - Semilinear parabolic system

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85044579803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044579803&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2018.01.003

DO - 10.1016/j.anihpc.2018.01.003

M3 - Article

AN - SCOPUS:85044579803

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

ER -