We consider nonnegative solutions to the exterior Dirichlet problem for quasilinear parabolic equations $u_t=\Delta u^m+u^p$ with $p=m+2/N$ and $m\geq 1$. In this paper we show that when $N\geq 3$ all nontrivial solutions to above problem blow up in finite time. For this aim, it is important to study the asymptotic behavior of solutions to the exterior Dirichlet problem for the quasilinear parabolic equations $u_t=\Delta u^m$.
"Critical Blow-Up for Quasilinear Parabolic Equations in Exterior Domains." Tokyo J. Math. 19 (2) 397 - 409, December 1996. https://doi.org/10.3836/tjm/1270042528