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Báo cáo hóa học: " Research Article Properties of Positive Solution for Nonlocal Reaction-Diffusion Equation with Nonlocal Boundary"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Properties of Positive Solution for Nonlocal Reaction-Diffusion Equation with Nonlocal Boundary | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 64579 12 pages doi 10.1155 2007 64579 Research Article Properties of Positive Solution for Nonlocal Reaction-Diffusion Equation with Nonlocal Boundary Yulan Wang Chunlai Mu and Zhaoyin Xiang Received 21 January 2007 Accepted 11 April 2007 Recommended by Robert Finn This paper considers the properties of positive solutions for a nonlocal equation with nonlocal boundary condition u x t JQ f x y u y t dy on dQ X 0 T . The conditions on the existence and nonexistence of global positive solutions are given. Moreover we establish the uniform blow-up estimates for the blow-up solution. Copyright 2007 Yulan Wang et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In this paper we consider the following nonlocal equation with nonlocal boundary condition ut Au J uq y t dy - kup x G Q t 0 u x t f x y u y t dy x G dQ t 0 Jn u x 0 u0 x x G Q 1.1 where p q 1 k 0 and Q G RN is a bounded domain with smooth boundary. The function f x y ị 0 is nonnegative continuous and defined for x G dQ y G Q while u0 is a nonnegative continuous function and satisfies the compatibility condition u0 x JQ f x y u0 y dy for x G dQ. Many physical phenomena were formulated into nonlocal mathematical models see 1-3 and studied by many authors. And in recent few years the reaction-diffusion equation with nonlocal source has been studied extensively. In particular M. Wang and 2 Boundary Value Problems Y. Wang 4 studied the heat equation with nonlocal source and local damping term ut - Au uq y t dy - kup 1.2 n which is subjected to homogeneous Dirichlet boundary condition. They concluded that the blowup occurs for large initial data if q p 1 while all solutions exist globally if 1 q p. In case of p q the issue depends on the comparison of n and k. .