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

Abstract : In this article we study the positive solutions of the parabolic semilinear system of competitive type \[ \left\{ \begin{array} [c]{c}% u_{t}-\Delta u+v^{p}=0,\\ v_{t}-\Delta v+u^{q}=0, \end{array} \right. \] in $\Omega\times\left( 0,T\right) $, where $\Omega$ is a domain of $\mathbb{R}^{N},$ and $p,q>0,$ $pq\neq1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form \[ u(x,t)\leqq Ct^{-(p+1)/(pq-1)},\qquad v(x,t)\leqq Ct^{-(q+1)/(pq-1)}% \] in $\omega\times\left( 0,T_{1}\right) ,$ for any domain $\omega \subset\subset\Omega$ and $T_{1}\in\left( 0,T\right) ,$ and $C=C(N,p,q,T_{1}% ,\omega).$ 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.
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Submitted on : Friday, February 11, 2011 - 3:47:29 PM
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  • HAL Id : hal-00465136, version 2
  • ARXIV : 1003.4189

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Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Cecilia Yarur. Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion. Advances in Nonlinear Studies, 2010, 10, pp.707-728. ⟨hal-00465136v2⟩

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