TY - JOUR

T1 - Construction and stability of blowup solutions for a non-variational semilinear parabolic system

AU - Ghoul, Tej Eddine

AU - Nguyen, Van Tien

AU - Zaag, Hatem

N1 - Funding Information:
H. Zaag is supported by the ERC Advanced Grant no. 291214, BLOWDISOL and by the ANR project ANAÉ ref. ANR-13-BS01-0010-03.
Funding Information:
H. Zaag is supported by the ERC Advanced Grant no. 291214, BLOWDISOL and by the ANR project ANA? ref. ANR-13-BS01-0010-03.
Publisher Copyright:
© 2018 Elsevier Masson SAS

PY - 2018/9

Y1 - 2018/9

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)(1+[Formula presented])]−[Formula presented],v(x,t)∼γ[(T−t)(1+[Formula presented])]−[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)(1+[Formula presented])]−[Formula presented],v(x,t)∼γ[(T−t)(1+[Formula presented])]−[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

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U2 - 10.1016/j.anihpc.2018.01.003

DO - 10.1016/j.anihpc.2018.01.003

M3 - Article

AN - SCOPUS:85044579803

SN - 0294-1449

VL - 35

SP - 1577

EP - 1630

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 6

ER -