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

TL;DR

This paper investigates the initial conditions necessary for solutions to a semilinear parabolic system, focusing on the initial trace and solvability based on initial data properties.

Contribution

It provides necessary conditions on initial data for the existence of solutions to a coupled semilinear parabolic system, analyzing initial trace properties.

Findings

Characterization of initial trace for solutions.

Necessary conditions on initial data for solvability.

Analysis of Radon measures and nonnegative functions as initial data.

Abstract

Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1\Delta u+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2\Delta v+u^q\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ u,v\ge 0 & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & \quad\mbox{in}\quad{\bf R}^N, \end{cases} \] 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 ${\bf R}^N$. 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).