A Cauchy problem of spatial-weighted reaction diffusion equations

This paper deals with a Cauchy problem of reaction-diffusion equations u(t) = Delta u + b(x)u(alpha)v(p) and v(t) = Delta v + a(x)u(q)v(beta) in R-N x [0, T) with p, q, alpha, beta >= 0. The weighted functions a(x), b(x) are locally Holder-continuous, satisfying that a(x) similar to vertical bar x vertical bar(m) and b(x) similar to vertical bar x vertical bar(n) as vertical bar x vertical bar -> + infinity for m, n >= 0. The complete classification on global and non-global solutions is obtained through determining the Fujita exponent which is affected by m, n, p, q, alpha, beta, N. The results of the paper extend the ones obtained by Escobedo and Herrero (1991) and Li, Sun and Zhang (2017), respectively, to the completely coupled equations. (C) 2019 Elsevier Ltd. All rights reserved.