Differential and Integral Equations

Nontrivial compact blow-up sets of lower dimension in a half-space

Mayte Pérez-Llanos and Julio D. Rossi

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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 $\partial {{\mathbb{R}}}^N_+ =\{ x_N =0\}$.

Article information

Differential Integral Equations, Volume 20, Number 11 (2007), 1211-1228.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions 35J60: Nonlinear elliptic equations 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations


Pérez-Llanos, Mayte; Rossi, Julio D. Nontrivial compact blow-up sets of lower dimension in a half-space. Differential Integral Equations 20 (2007), no. 11, 1211--1228. https://projecteuclid.org/euclid.die/1356039285

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