Stochastic generalized porous media equations driven by Lévy noise with increasing Lipschitz nonlinearities

Abstract

We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by Lévy noise on a $\sigma$-finite measure space $(E,{\mathcal {B}}(E),\mu )$, and with the Laplacian replaced by a negative definite self-adjoint operator. The coefficient $\Psi$ is only assumed to satisfy the increasing Lipschitz nonlinearity assumption, without the restriction $r\Psi (r)\rightarrow \infty$ as $r\rightarrow \infty$ for $L^2(\mu )$-initial data. We also extend the state space, which avoids the transience assumption on L or the boundedness of $L^{-1}$ in $L^{r+1}(E,{\mathcal {B}}(E),\mu )$ for some $r\ge 1$. Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e., $L=-(-\Delta )^\alpha ,\ \alpha \in (0,1]$, generalized $\mathrm Schr\ddot{o}dinger$ operators, i.e., $L=\Delta +2\frac{\nabla \rho }{\rho }\cdot \nabla$, and Laplacians on fractals.