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

Abstract

Let (𝑢, 𝑣) be a solution to a semilinear parabolic system

(P)    \begin{cases}𝜕_𝑡 𝑢 = 𝐷₁ Δ 𝑢+𝑣^𝑝 in 𝐑^𝑁 ×(0, 𝑇),   𝜕_𝑡 𝑣 = 𝐷₂ Δ 𝑣+𝑢^𝑞 in 𝐑^𝑁 ×(0, 𝑇),   𝑢, 𝑣 ≥ 0 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 Radon measures or nonnegative measurable functions in 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).