Nontrivial compact blow-up sets of lower dimension in a half-space. (English) Zbl 1212.35017

Pérez-Llanos, Mayte; Rossi, Julio D.

Differ. Integral Equ. 20, No. 11, 1211-1228 (2007).

Summary: In this paper we provide examples of blowing-up solutions to parabolic problems in a half space, \(\mathbb R^N_+\times \mathbb R^M=\{x_N>0\}\times \mathbb R^M\), with nontrivial blow-up sets of dimension strictly smaller than the space dimension. To this end we prove existence of a nontrivial compactly supported solution to \(\nabla (| \nabla \varphi | ^{p-2}\nabla \varphi )=\varphi \) in the half space \(\mathbb R^N_+=\{x_N>0\}\) with the nonlinear boundary condition \(-| \nabla \varphi | ^{p-2}\frac {\partial \varphi }{\partial x_N}=\varphi ^{p-1}\) on \(\partial \mathbb R^N_+=\{x_N=0\}\).

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35K65	Degenerate parabolic equations
35J25	Boundary value problems for second-order elliptic equations
35J60	Nonlinear elliptic equations

Keywords:

parabolic problem; half space; blowing-up solution