# Pointwise estimates and existence of solutions of porous medium and p-Laplace evolution equations with absorption and measure data

created by nguyen on 22 Jul 2018

[BibTeX]

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 $p>2$.