An example of blowup for a degenerate parabolic equation with a nonlinear boundary condition

Abstract

In this paper, a nonlinear parabolic equation of the form $u_t=(a(u_x))_x$ for $x\in(0,1),\ t>0,\ a(u_x)=|u_x|^{p-2}u_x$ if $u_x\geq\eta>0,\ 1<p<2$, with nonlinear boundary condition $a(u_x(1,t))=|u|^{q-2}u(1,t)$, is considered. It is proved that if $qp-3p+2>0$, then the solutions blow up in finite time. Moreover, estimates on the blow-up profile (in $x$) and the blow-up rate (in $t$) for $x=1$ are derived.