Existence of solutions for a semilinear parabolic system with singular initial data

TL;DR

This paper establishes sharp conditions on initial data for the existence of solutions to a semilinear parabolic system with singular initial data, using advanced function spaces.

Contribution

It introduces new criteria involving uniformly local Morrey and Zygmund spaces for solution existence with singular initial data.

Findings

Derived sharp existence conditions for solutions

Utilized uniformly local Morrey spaces in analysis

Applied uniformly local weak Zygmund spaces for initial data

Abstract

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] 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 nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.