Evolution equations of p-Laplace type with absorption or source terms and measure data

Abstract

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=Ω\times(0,T).$ We consider problems\textit{ }of the type % \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u\pm\mathcal{G}(u)=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where ${Δ_{p}}$ is the $p$-Laplacian, $μ$ is a bounded Radon measure, $u_{0}\in L^{1}(Ω),$ and $\pm\mathcal{G}(u)$ is an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$ or $\mathcal{G}$ has an exponential type. We prove the existence of renormalized solutions for any measure $μ$ in the subcritical case, and give sufficient conditions for existence in the general case, when $μ$ is good in time and satisfies suitable capacitary conditions.