Well-posedness for a pseudomonotone evolution problem with multiplicative noise

Abstract

Our aim is the study of well-posedness for the stochastic evolution equation

$$\begin{aligned} \mathrm{d}u -\mathrm {div}\,(|\nabla u|^{p-2}\nabla u +F(u)) \,\mathrm{d} t = H(u) \,\mathrm{d}W, \end{aligned}$$

for $T>0$, on a bounded Lipschitz domain $D \subset \mathbb {R}^d$ with homogeneous Dirichlet boundary conditions, initial values $u_0\in L^2(D)$, $p>2$ and $F:\mathbb {R}\rightarrow \mathbb {R}^d$ Lipschitz continuous. W(t) is a cylindrical Wiener process in $L^2(D)$ with respect to a filtration $(\mathcal {F}_t)$ satisfying the usual assumptions on a complete, countably generated probability space $(\varOmega ,\mathcal {F},\mathbb {P})$. We consider the case of multiplicative noise satisfying appropriate regularity conditions. By a semi-implicit time discretization, we obtain approximate solutions. Using the theorems of Skorokhod and Prokhorov, we are able to pass to the limit and show existence of martingale solutions. We establish pathwise $L^1$-contraction and uniqueness and obtain existence and uniqueness of strong solutions.