Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form dXt−divγ(∇Xt)dt+β(Xt)dt∋B(t,Xt)dWt, where γ and β are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on Rd and R respectively, and W is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray–Lions conditions on γ and with no restrictive smoothness or growth assumptions on β. The operator B is assumed to be Hilbert–Schmidt and to satisfy some classical Lipschitz conditions in the second variable.