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

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 \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta }(|u|^{m-1}u)+|u|^{q-1}u=\mu ~ \text{in }\Omega \times (0,T), \\ {u}=0~~~\text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma , \end{array} \right. \end{equation*} 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 \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta _{p}}u+|u|^{q-1}u=\mu ~~\text{in }\Omega \times (0,T), \\ {u}=0 ~ \text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma . \end{array} \right. \end{equation*} where $q>p-1$ and $p>2$.