Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion

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1 LMPT - Laboratoire de Mathématiques et Physique Théorique 2 Departamento de Matematicas PUC 3 Departamento de Matematicas y CC - Departamento de Matematicas y CC

Abstract : In this article we study the positive solutions of the parabolic semilinear system of competitive type \ \left\{ \begin{array} c{c}% u {t}-\Delta u+v^{p}=0,\\ v {t}-\Delta v+u^{q}=0, \end{array} ight. \ in $\Omega\times\left 0,T ight$, where $\Omega$ is a domain of $\mathbb{R}^{N},$ and $p,q>0,$ $pq eq1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form \ ux,t\leqq Ct^{-p+1-pq-1},\qquad vx,t\leqq Ct^{-q+1-pq-1}% \ in $\omega\times\left 0,T {1} ight ,$ for any domain $\omega \subset\subset\Omega$ and $T {1}\in\left 0,T ight ,$ and $C=CN,p,q,T {1}% ,\omega.$ For $p,q>1,$ we prove the existence of an initial trace at time 0, which is a Borel measure on $\Omega.$ Finally we prove that the punctual singularities at time $0$ are removable when \$p,q\geqq1+2-N.

Keywords : singularities Parabolic semilinear systems of reaction-diffusion competitive systems backward estimates initial trace singularities.

Autor: Marie-Françoise Bidaut-Véron - Marta Garcia-Huidobro - Cecilia Yarur -

Fuente: https://hal.archives-ouvertes.fr/

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