Calculus of Variations and Geometric Measure Theory

M. F. Bidaut-VĂ©ron - Q. H. Nguyen

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

created by nguyen on 22 Jul 2018


Published Paper

Inserted: 22 jul 2018
Last Updated: 22 jul 2018

Journal: Communications in Contemporary Mathematics
Year: 2015


Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We consider problems\textit{ }of the type $u_{t}-\Delta_{p}u\pm\mathcal{G}(u)=\mu$ in $Q$, $u=0$ on$\partial\Omega\times(0,T)$, $u(0)=u_{0}$ in $\Omega$, where ${\Delta_{p}}$ is the $p$-Laplacian, $\mu$ is a bounded Radon measure, $u_{0}\in L^{1}(\Omega),$ 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 $\mu$ in the subcritical case, and give sufficient conditions for existence in the general case, when $\mu$ is good in time and satisfies suitable capacitary conditions.