Optimal singularities of initial functions for solvability of a semilinear parabolic system

Abstract

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

(P)\begin{cases}𝜕_𝑡 𝑢 = 𝐷₁ Δ 𝑢 + 𝑣^𝑝,𝑥 ∈ 𝐑^𝑁, 𝑡 > 0,𝜕_𝑡 𝑣= 𝐷₂ Δ 𝑣 + 𝑢^𝑞,𝑥 ∈ 𝐑^𝑁, 𝑡 > 0,(𝑢(⋅,0), 𝑣(⋅,0))= (𝜇, 𝜈), 𝑥 ∈ 𝐑^𝑁,\end{cases}

where 𝐷₁, 𝐷₂ > 0, 0 < 𝑝 ≤ 𝑞 with 𝑝𝑞 > 1 and (𝜇, 𝜈) is a pair of nonnegative Radon measures or nonnegative measurable functions in 𝐑^𝑁. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.