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

Marie-Françoise Bidaut-Véron (1) , Marta Garcia-Huidobro (2) , Cecilia Yarur (3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 Departamento de Matemáticas [Santiago de Chile]
3 Departamento de Matematicas y CC - Departamento de Matematicas y CC

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.

Mots clés: Parabolic semilinear systems of reaction-diffusion; competitive systems; backward estimates; initial trace; singularities.; singularities

Domaines: Equations aux dérivées partielles [math.AP]

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Cecilia Yarur. Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion. 2010. ⟨hal-00465136v1⟩