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

TL;DR

This paper investigates positive solutions of a reaction-diffusion parabolic system, establishing local upper bounds in superlinear cases, existence of initial traces as measures, and conditions for removable singularities at initial time.

Contribution

It provides new backward blow-up estimates, proves the existence of initial traces as Borel measures, and characterizes removable singularities for the system.

Findings

Established local upper estimates for solutions in superlinear case.

Proved existence of initial trace as a Borel measure.

Identified conditions under which initial singularities are removable.

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.