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-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$

$u(x)=0$ on $\partial \Omega,$

where $-\infty<\gamma<\frac{N-p}{p}$, $\Omega$ is a bounded domain in $\mathbb R^N$ such that $0\in\Omega$ and $f(x,u)$ is a Caratheodory function under suitable conditions that will be stated in each section.

If $1 < p < 2$ and $0 < q < p-1$, there is no finite extinction time.

If $p > 2$ and $0 < q< 1$, there is no finite speed of propagation.

In both cases the result is optimal.

$u_t-$ 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 $\gamma+1>0$.

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