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 ∂tu=Δu+g(v)inRN×(0,T),∂tv=Δv+f(u)inRN×(0,T),(u(x,0),v(x,0))=(u0(x),v0(x))inRN,where f(u) and g(v) grow rapidly, u0 and v0 are possibly unbounded nonnegative initial functions in RN (N≥1) and T is a positive constant. A typical example is (f(u),g(v))=(eup,evq), p≥1 and q≥1. We show that if (u0,v0) satisfies a certain integrability condition, then the local in time solution exists. Moreover, we show that there exists (u0,v0) not satisfying the integrability condition such that the solution does not exist.