Weakly coupled reaction-diffusion systems with rapidly growing nonlinearities and singular initial data

We study existence and nonexistence of a local in time solution for the weakly coupled reaction-diffusion system {partial derivative(t)u = Delta u + g(v) in R-N x (0, T), partial derivative(t)v = Delta v + f(u) in R-N x (0, T), (u(x, 0), v(x,0)) = (u(0)(x), v(0)(x)) in R-N, where f (u) and g(v) grow rapidly, u(0) and v(0) are possibly unbounded nonnegative initial functions in R-N (N >= 1) and T is a positive constant. A typical example is (f (u), g(v)) = (e(up), e(vq)), p >= 1 and q >= 1. We show that if (u(0), v(0)) satisfies a certain integrability condition, then the local in time solution exists. Moreover, we show that there exists (u(0), v(0)) not satisfying the integrability condition such that the solution does not exist. (C) 2019 Elsevier Ltd. All rights reserved.