A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, [Formula: see text] and [Formula: see text], we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

In this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.