Blow up and quenching for a problem with nonlinear boundary conditions

Abstract

In this article, we study the blow up behavior of the heat equation $ u_t=u_{xx}$ with $u_x(0,t)=u^{p}(0,t)$, $u_x(a,t)=u^q(a,t)$. We also study the quenching behavior of the nonlinear parabolic equation $v_t=v_{xx}+2v_x^{2}/(1-v)$ with $v_x(0,t)=(1-v(0,t))^{-p+2}$, $ v_x(a,t)=(1-v(a,t))^{-q+2}$. In the blow up problem, if $u_0$ is a lower solution then we get the blow up occurs in a finite time at the boundary $x=a$ and using positive steady state we give criteria for blow up and non-blow up. In the quenching problem, we show that the only quenching point is $x=a$ and $v_t$ blows up at the quenching time, under certain conditions and using positive steady state we give criteria for quenching and non-quenching. These analysis is based on the equivalence between the blow up and the quenching for these two equations.

Keywords

• Heat equation
• nonlinear parabolic equation
• blow up
• nonlinear boundary condition
• quenching
• maximum principle