*Published Paper*

**Inserted:** 22 jul 2018

**Last Updated:** 22 jul 2018

**Journal:** Communications in Contemporary Mathematics

**Year:** 2015

**Doi:** https://www.worldscientific.com/doi/abs/10.1142/S0219199715500066

**Abstract:**

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.

**Download:**