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

2019/10/15 by Yohei Fujishima, Kazuhiro Ishige, Fujishima, Yohei +1 · 2 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.1910.06546

Abstract

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

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• Initial traces and solvability of the fast diffusion equation with power-type nonlinearity
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