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

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  • 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 $\gamma+1>0$.

    Mathematics Subject Classification: 35K10, 35K15, 35K65, 35J70, 46E35.


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