Annali di Matematica Pura ed Applicata

, Volume 165, Issue 1, pp 315–336

# A semilinear parabolic system in a bounded domain

• M. Escobedo
• M. A. Herrero
Article

DOI: 10.1007/BF01765854

Escobedo, M. & Herrero, M.A. Annali di Matematica pura ed applicata (1993) 165: 315. doi:10.1007/BF01765854

## Summary

Consider the system
$$\left\{ \begin{gathered} u_t - \Delta u = v^p , in Q = \{ (t, x), t > 0, x \in \Omega \} , \hfill \\ v_t - \Delta v = u^q , in Q , \hfill \\ u(0, x) = u_0 (x) v(0, x) = v_0 (x) in \Omega , \hfill \\ u(t, x) = v(t, x) = 0 , when t \geqslant 0, x \in \partial \Omega , \hfill \\ \end{gathered} \right.$$
(S)
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).