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

作者：Scarpa Luca^*

来源：Journal of Differential Equations, 2017, 263(4): 2113-2156.

DOI：10.1016/j.jde.2017.03.041

• Doubly nonlinear stochastic equation
• Divergence
• Variational approach
• Existence of solutions
• Continuous dependence
• Multiplicative noise

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form dX(t) - div gamma (del X-t) d(t) + beta(X-t) dt (sic) B(t, X-t) dW(t), where gamma and beta are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on R-d 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 y and with no restrictive smoothness or growth assumptions on beta. The operator B is assumed to be Hilbert Schmidt and to satisfy some classical Lipschitz conditions in the second variable.

• 出版日期2017-8-15