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

  • Tej-Eddine Ghoul , Van Tien Nguyen and Hatem Zaag EMAIL logo
Published/Copyright: February 15, 2019

Abstract

In this note, we consider the semilinear heat system

tu=Δu+f(v),tv=μΔv+g(u),μ>0,

where the nonlinearity has no gradient structure taking of the particular form

f(v)=v|v|p-1andg(u)=u|u|q-1with p,q>1,

or

f(v)=epvandg(u)=equwith p,q>0.

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, 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, 3, 381–450].

Award Identifier / Grant number: ANR-13-BS01-0010-03

Funding statement: H. Zaag is supported by the ANR project ANAÉ ref. ANR-13-BS01-0010-03.

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Received: 2018-10-14
Revised: 2019-01-13
Accepted: 2019-01-14
Published Online: 2019-02-15
Published in Print: 2019-10-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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