Large Time Behavior and Uniqueness of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

Abstract

where N ≥ 1, N∗ = {1, 2, . . . N}, d ≥ 1, pi > 0(i ∈ N∗) and ui,0(i ∈ N∗) are nonnegative bounded and continuous functions. Throughout this paper we mean uN+i = ui, uN+i,0 = ui,0, pN+i = pi for each i ∈ Z and u = (u1, u2, · · · , uN), u0 = (u1,0, u2,0, · · · , uN,0). Problem (1) has a nonnegative and bounded solution at least locally in time (see Theorem 2.1). For any given initial value u0, let T ∗ = T ∗(u0) be the maximal existence time of the solution. If T ∗ = ∞, it is called a global solution. On the other hand, if T ∗ < ∞, there exists i ∈ N∗ such that