Initial traces and solvability of Cauchy problem to a semilinear parabolic system

Abstract

Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1Δu+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2Δv+u^q\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ u,v\ge 0 & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & \quad\mbox{in}\quad{\bf R}^N, \end{cases} \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(\mu ,\nu )$ is a pair of Radon measures or nonnegative measurable functions in ${\mathbf{R}}^N$. In this paper we study qualitative properties of the initial trace of the solution $(u,v)$ and obtain necessary conditions on the initial data $(\mu ,\nu )$ for the existence of solutions to problem (P).

References in corpus (2)

• Supersolutions for a class of nonlinear parabolic systems
• Optimal singularities of initial functions for solvability of a semilinear parabolic system