Quenching problems for a semilinear reaction-diffusion system with singular boundary outflux

Abstract

In this paper, we study two quenching problems for the following semilinear reaction-diffusion system:
$$u_t=u_{xx}+(1-v)^{-p_1}, 0<x<1,~ 0<t<T,$$
$$v_t=v_{xx}+(1-u)^{-p_2}, 0<x<1,~ 0<t<T,$$
$$u_x (0,t) =0,~ u_x (1,t) =-v^{-q_1}(1,t),~ 0<t<T,$$
$$v_x (0,t) =0,~ v_x (1,t) =-u^{-q_2}(1,t),~ 0<t<T,$$
$$u(x,0) =u_0 (x) <1, v( x,0)=v_0 ( x) <1,~  0\le x\le 1,$$
where $p_1, p_2, q_1, q_2$ are positive constants and $u_0 (x), v_0 (x)$ are positive smooth functions.  We firstly get a local exisence result for this system.  In the first problem, we show that quenching occurs in finite time, the only quenching point is $x=0$ and $(u_t ,v_t )$ blows up at the quenching time under the certain conditions.  In the second problem, we show that quenching occurs in finite time, the only quenching point is $x=1$ and $(u_t ,v_t )$ blows up at the quenching time under the certain conditions.