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March  2006, 5(1): 29-54. doi: 10.3934/cpaa.2006.5.29

Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid

2. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 MADRID, Spain

Received  January 2005 Revised  September 2005 Published  December 2005

We study the following parabolic problem

$u_t-$ div $(|x|^{-p\gamma}|\nabla u|^{p-2}\nabla u) = \lambda f(x,u), u\ge 0$ in $\Omega\times (0,T)$,

$ B(u) = 0$ on $\partial\Omega\times (0,T),$

$ u(x,0) = \varphi (x)\quad$ if $x\in\Omega$,

where $\Omega\subset\mathbb R^N$ is a smooth bounded domain with $0\in\Omega$,

$B(u)\equiv u\chi_{\Sigma_1\times(0,T)}+|x|^{-p\gamma} |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\chi_{\Sigma_2 \times (0,T)}$

and $-\infty<\gamma<\frac{N-p}{p}$. The boundary conditions over $\partial\Omega\times (0,T)$ verify hypotheses that will be precised in each case.
Mainly, we will consider the second member $f(x,u)=\frac{u^{\alpha}}{|x|^{p(\gamma+1)}}$ with $ \alpha\ge p-1$, as a model case. The main points under analysis are some existence, nonexistence and complete blow-up results related to some Hardy-Sobolev inequalities and a weak version of Harnack inequality, that holds for $p\ge 2$ and $\gamma+1>0$.

Citation: B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure and Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29
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