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

Abstract

Let (𝑢, 𝑣) be a solution to the Cauchy problem for a semilinear parabolic system

(P) \begin{cases} 𝜕_𝑡𝑢 = 𝐷₁ Δ𝑢 + 𝑣^𝑝 in ℝ^𝑁 × (0, 𝑇), 𝜕_𝑡𝑣 = 𝐷₂ Δ𝑣 + 𝑢^𝑞 in ℝ^𝑁 × (0, 𝑇), (𝑢(⋅,0), 𝑣(⋅,0)) = (𝜇, 𝜈) in ℝ^𝑁, \end{cases}

where 𝑁 ≥ 1, 𝑇 > 0, 𝐷₁ > 0, 𝐷₂ > 0, 0 < 𝑝 ≤ 𝑞 with 𝑝𝑞 > 1, and (𝜇, 𝜈) is a pair of nonnegative Radon measures or locally integrable nonnegative functions in 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.