Calculus of Variations and Geometric Measure Theory

M. F. Bidaut-VĂ©ron - Q. H. Nguyen

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


Published Paper

Inserted: 22 jul 2018
Last Updated: 22 jul 2018

Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci.
Year: 2016


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