| Abstract: | In this paper, we study two quenching problems for the following semilinear reaction-diffusion system:
ut=uxx+(1−v)−p1,0<x<1, 0<t<T,ut=uxx+(1−v)−p1,0<x<1, 0<t<T,
vt=vxx+(1−u)−p2,0<x<1, 0<t<T,vt=vxx+(1−u)−p2,0<x<1, 0<t<T,
ux(0,t)=0, ux(1,t)=−v−q1(1,t), 0<t<T,ux(0,t)=0, ux(1,t)=−v−q1(1,t), 0<t<T,
vx(0,t)=0, vx(1,t)=−u−q2(1,t), 0<t<T,vx(0,t)=0, vx(1,t)=−u−q2(1,t), 0<t<T,
u(x,0)=u0(x)<1,v(x,0)=v0(x)<1, 0≤x≤1,u(x,0)=u0(x)<1,v(x,0)=v0(x)<1, 0≤x≤1,
where p1,p2,q1,q2p1,p2,q1,q2 are positive constants and u0(x),v0(x)u0(x),v0(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=0x=0 and (ut,vt)(ut,vt) 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=1x=1 and (ut,vt)(ut,vt)blows up at the quenching time under the certain conditions. |
