Stochastic generalized porous media equations driven by L\'{e}vy noise with increasing Lipschitz nonlinearities

TL;DR

This paper proves the existence and uniqueness of solutions to stochastic porous media equations driven by Lévy noise, accommodating increasing Lipschitz nonlinearities and extending the state space to include fractional Laplacians, Schrödinger operators, and fractal Laplacians.

Contribution

It establishes strong solution existence and uniqueness for a broad class of stochastic porous media equations with minimal restrictions on nonlinearities and operator properties.

Findings

Proved strong solutions exist and are unique under new conditions.

Extended the framework to include fractional Laplacians and Schrödinger operators.

Handled equations on fractals and with less restrictive nonlinear growth conditions.

Abstract

We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by L\'{e}vy 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 $r\geq1$. Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e. $L=-(-\Delta)^\alpha,\ \alpha\in(0,1]$, generalized Schr\"{o}dinger operators, i.e. $L=\Delta+2\frac{\nabla \rho}{\rho}\cdot\nabla$, and Laplacians on fractals.