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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

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