Nonlinear diffusion equations with and without memory and stochastic perturbation: theory and numerical approximation

In the dissertation, we deal with the existence of solutions to a nonlinear diffusion equation with memory and L1-data, to a nonlinear diffusion equation with Hölder continuous stochastic perturbation, and, additionally, we prove convergence of a finite-volume scheme for a stochastic heat equation with a nonlinear multiplicative noise. In the first part, we show the existence of entropy solutions for a time-fractional porous medium type equation with L1-data, where the porous medium operator depends on the time. Therefore, we approximate the L1-data by bounded functions and use a known existence result of weak solutions for bounded data. Additionally, we apply a known contraction principle for weak solutions, which can be adopted to the entropy solutions. In the second part, we show the existence and uniqueness of probabilistically strong solutions to a pseudomonotone stochastic evolution equation with a Hölder continuous multiplicative noise term. In order to show the well-posedness, we simultaneously regularize the Hölder noise term by inf-convolution and add a perturbation by a higher order operator to the equation. Using a stochastic compactness argument, we may pass to the limit and we obtain first a martingale solution. Then, by a pathwise uniqueness argument we get existence of a probabilistically strong solution. In the third part, we consider a finite-volume approximation for a heat equation with a nonlinear multiplicative noise. To be more precise, we use a two-point flux approximation in space and a semi-implicit discretization in time. By using a stochastic compactness argument based on the theorems of Prokhorov and Skorokhod, we show convergence of the scheme to a martingale solution. Because we can show the pathwise uniqueness of solutions to our parabolic equation, an argument of Gyöngy and Krylov provides convergence of the finite-volume approximations to the unique variational solution of the stochastic heat equation.