This paper considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the blow-up rate estimate for the blow-up solution. 1. Introduction This paper studies the singularity properties of the following reaction-diffusion system with nonlocal boundary condition: where is a bounded domain of , , with smooth boundary and is the closure of . , and are positive numbers which ensure that the equations in (1) are completely coupled with the nonlinear reaction terms. The functions defined for , are nonnegative and continuous. The initial values and are nonnegative, which are mathematically convenient and currently followed throughout the paper. We also assume that satisfies the compatibility condition on , and that and for any for the sake of the meaning of nonlocal boundary. Denote that , , and , where . A pair of functions is called a classical solution of problem (1) if for some , , and satisfies (1). The local existence of classical solution of (1) is standard (see [1, 2]). If , it is easy to see , and we say that the solution of problem (1) blows up at finite time . If , is called a global solution of problem (1). Over the past few years, many physical phenomena were formulated as nonlocal mathematical models (see [3, 4]). It has also being suggested that nonlocal growth terms present a more realistic model in physics for compressible reactive gases. Problem (1) arises in the study of the heat transfer with local source (see [5, 6]) and in the study of population dynamics (see [7, 8]). In recent years many authors have investigated the following initial boundary value problem of reaction-diffusion system: with Dirichlet, Neumanns or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see [2, 4, 9–23] and the literatures cited therein). Specially, when have the form a classical result is (see [9, 10, 12, 20]). Theorem A. The system (2) ( is of the form (3)) with homogenous Dirichlet boundary condition admits a unique global solution for any nonnegative initial data if and only if , , and . However, there are some important phenomena formulated as parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [24–26]). In this case, the solution could be used to describe the entropy per volume of the material. The problem of nonlocal boundary conditions for linear
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