Construction and stability of type i blowup solutions for non-variational semilinear parabolic systems

Tej Eddine Ghoul, Van Tien Nguyen, Hatem Zaag

Research output: Contribution to journalArticlepeer-review

Abstract

In this note, we consider the semilinear heat system (equation presented) where the nonlinearity has no gradient structure taking of the particular form (equation presented) We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: The reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 6, 1577-1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 (1992), no. 3, 381-450].

Original languageEnglish (US)
Pages (from-to)299-312
Number of pages14
JournalAdvances in Pure and Applied Mathematics
Volume10
Issue number4
DOIs
StatePublished - Oct 1 2019

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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