A semilinear parabolic system in a bounded domain

Abstract

Consider the system (S) {ut–Δu=v(p),inQ={(t,x),t>0, x∈Ω}, vt–Δv=u(q), inQ, u(0,x)=u0(x)v(0,x)=v0(x)inΩ, u(t,x)=v(t,x)=0, whent≥0, x∈∂Ω,
where Ω is a bounded open domain in ℝN with smooth boundary, p and q are positive parameters, and functions u0 (x), v0(x) are continuous, nonnegative and bounded. It is easy to show that (S) has a nonnegative classical solution defined in some cylinder QT=(0,T)×Ω with T||∞. We prove here that solutions are actually unique if pq||1, or if one of the initial functions u0, v0 is different from zero when 0<pq<1. In this last case, we characterize the whole set of solutions emanating from the initial value (u0, v0)=(0,0). Every solution exists for all times if 0<pq| |1, but if pq>1, solutions may be global or blow up in finite time, according to the size of the initial value (u0,v0).