Global blow-up for a heat system with localized sources and absorptions

Abstract

In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic system

$$\begin{array}{*{20}c} {u_t = \Delta u + v^p (x_0 ,t) - au^r , x \in \Omega , t > 0,} \\ {v_t = \Delta v + u^q (x_0 ,t) - bv^s , x \in \Omega , t > 0,} \\ \end{array} $$

subject to homogeneous Dirichlet conditions and nonnegative initial data, where x ₀ ∈ Ω is a fixed point, p, q, r, s ≥ 1 and a, b > 0 are constants. In the situation when nonnegative solution (u, υ) of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1,

$$\begin{array}{*{20}c} {\mathop {\lim }\limits_{t \to T*} (T* - t)^{\frac{{p + 1}}{{pq - 1}}} u(x,t) = (p + 1)^{\frac{1}{{pq - 1}}} (q + 1)^{\frac{p}{{pq - 1}}} (pq - 1)^{ - \frac{{p + 1}}{{pq - 1}}} ,} \\ {\mathop {\lim }\limits_{t \to T*} (T* - t)^{\frac{{q + 1}}{{pq - 1}}} v(x,t) = (p + 1)^{\frac{q}{{pq - 1}}} (q + 1)^{\frac{1}{{pq - 1}}} (pq - 1)^{ - \frac{{q + 1}}{{pq - 1}}} ,} \\ \end{array} $$

are obtained uniformly on compact subsets of Ω, where T* is the blow-up time.