Blowup solutions for a reaction–diffusion system with exponential nonlinearities

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+epv,∂tv=μΔv+equ,u(⋅,0)=u0,v(⋅,0)=v0,p,q,μ>0, in the whole space RN. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. 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. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

Original languageEnglish (US)
Pages (from-to)7523-7579
Number of pages57
JournalJournal of Differential Equations
Volume264
Issue number12
DOIs
StatePublished - Jun 15 2018

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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