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 $u_x\ge \eta >0,1<p<2$, with nonlinear boundary condition $a(u_{x}r(1,t))=|u{|}^{q–2}u(l,t)$ is considered. It is proved that if $qp-3p+2>0$, then the solutions blow up in finite time. Moreover, estimates on the blowup profile (in $x$) and the blowup rate (in $t$) for $x=1$ are derived.