A critical exponent of Fujita type for a nonlinear reaction-diffusion system on a Riemannian manifold

作者：Ru, Qiang^*

来源：Journal of Mathematical Analysis and Applications, 2013, 401(2): 488-500.

DOI：10.1016/j.jmaa.2012.12.021

• Critical exponent
• Reaction-diffusion system
• Riemannian manifold

In this paper, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction-diffusion system @@@ {u(t) - Delta u = W (x)v(p) + S(x) in M-n x (0, infinity), @@@ v(t) - Delta v = F(x)u(d) + G(x) in M-n x (0, infinity), @@@ u(x, 0) = u(0) (x) in M-n, @@@ v(x, 0) = v(0)(x) in M-n, @@@ where M-n (n >= 3) is a non-compact complete Riemannian manifold, d is the Laplace-Beltrami operator, and S(x), G(x) are non-negative L-loc(1) functions. We assume that both u(0)(x) and v(0)(x) are non-negative, smooth and bounded functions, and constants p, d > 1. When p = d, there is an exponent p* which is critical in the following sense. When p is an element of (1, p*] the above problem has no global positive solution for any non-negative constants S(x), G(x) not identically zero. When p is an element of [p*, infinity), the problem has a global positive solution for some S(x), G(x) > 0 and u(0)(x), v(0)(x) >= 0.

• 出版日期2013-5
• 单位浙江大学