Existence and nonexistence of global solutions for a semilinear reaction-diffusion system

This paper is concerned with the blow-up and global existence of nonnegative solutions to the following Cauchy problem u(t) - Delta u = v(p), t > 0, x is an element of R-N, v(t) - Delta v = a(x)u(q), t > 0, x is an element of R-N, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), x is an element of R-N, where the constants p, q > 0 and a(x) (sic) 0 is on the order vertical bar x vertical bar(m) as vertical bar x vertical bar -> infinity, m is an element of R. The Fujita critical exponent is determined when m >= 0, and some results of global existence of solution under some assumptions when m < 0 are also obtained. The results extend those in Escobedo and Herrero (1991) [9] and indicate that m affects the Fujita critical exponent. (C) 2016 Elsevier Inc. All rights reserved.