Quenching behavior of semilinear heat equations with singular boundary conditions

Abstract

In this article, we study the quenching behavior of solution to the semilinear heat equation $$ v_t=v_{xx}+f(v), $$ with $f(v)=-v^{-r}$ or $(1-v)^{-r}$ and $$ v_x(0,t)=v^{-p}(0,t), \quad v_x(a,t) =(1-v(a,t))^{-q}. $$ For this, we utilize the quenching problem $u_t=u_{xx}$ with $u_x(0,t)=u^{-p}(0,t)$, $u_x(a,t)=(1-u(a,t))^{-q}$. In the second problem, if $u_0$ is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is $x=0$ ($x=a$) and $u_t$ blows up at quenching time. Further, we obtain a local solution by using positive steady state. In the first problem, we first obtain a local solution by using monotone iterations. Finally, for $f(v)=-v^{-r}$ ($(1-v)^{-r}$), if $v_0$ is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is $x=0$ ($x=a$) and $v_t$ blows up at quenching time.

Keywords

• Heat equation
• singular boundary condition
• quenching
• maximum principle
• monotone iteration