*Published Paper*

**Inserted:** 22 jul 2018

**Last Updated:** 22 jul 2018

**Journal:** Ann. Sc. Norm. Super. Pisa Cl. Sci.

**Year:** 2016

**Doi:** http://annaliscienze.sns.it/public/pdf/abstracts/2016/Abstract_Bidaut-V%C3%A9ron%20et%20al.pdf

**Abstract:**

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}(N\geq 2)$. We obtain a
necessary and a sufficient condition, expressed in terms of capacities, for
existence of a solution to the porous medium equation with absorption
$u_{t}-\Delta(

u

^{m-1}u)+

u

^{q-1}u=\mu$ in
$\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$,
$u(0)=\sigma$ in $\Omega$,
where $\sigma $ and $\mu $ are bounded Radon measures, $q>\max (m,1)$, $m>%
\frac{N-2}{N}$. We also obtain a sufficient condition for existence of a
solution to the $p$-Laplace evolution equation
$u_{t}-\Delta_p u+

u

^{q-1}u=\mu$ in
$\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$,
$u(0)=\sigma$ in $\Omega$,
where $q>p-1$ and $p>2$.

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