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

2024/07/03 by Fujishima, Yohei, Ishige, Kazuhiro, Kawakami, Tatsuki
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.02847

Abstract

Let (u,v) be a solution to the Cauchy problem for a semilinear parabolic system (P) \cases ∂_t u=D₁Δu+v^p amp; \quadin ℝ^N×(0,T),
∂_t v=D₂Δv+u^q amp; \quadin ℝ^N×(0,T),
(u(⋅,0),v(⋅,0))=(μ,ν) amp; \quadin ℝ^N, where N≥ 1, T>0, D₁>0, D₂>0, 01, and (μ,ν) 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.

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