Abstract:
This thesis consists of two main parts.
The rst part concerns the existence of weak probabilistic solutions (called elsewhere
martingale solutions) for a stochastic quasilinear parabolic equation of generalized
polytropic ltration, characterized by the presence of a nonlinear elliptic part admitting
nonstandard growth.
The deterministic version of the equation was rst introduced and studied by Samokhin
in [178] as a generalized model for polytropic ltration. Our objective is to investigate
the corresponding stochastic counterpart in the functional setting of generalized
Lebesgue and Sobolev spaces. We establish an existence result of weak probabilistic
solutions when the forcing terms do not satisfy Lipschitz conditions and the noise
involves cylindrical Wiener processes.
The second part is devoted to the existence and uniqueness results for a class of
strongly nonlinear stochastic parabolic partial di erential equations.
This part aims to treat an important class of higher-order stochastic quasilinear
parabolic equations involving unbounded perturbation of zeroth order. The deterministic
case was studied by Brezis and Browder (Proc. Natl. Acad. Sci. USA,
76(1): 38-40, 1979). Our main goal is to provide a detailed study of the corresponding
stochastic problem. We establish the existence of a probabilistic weak solution
and a unique strong probabilistic solution. The main tools used in this part of the
thesis are a regularization through a truncation procedure which enables us to adapt
the work of Krylov and Rozosvkii (Journal of Soviet Mathematics, 14: 1233-1277,
1981), combined with analytic and probabilistic compactness results (Prokhorov and
Skorokhod Theorems), the theory of pseudomonotone operators, and a Banach space
version of Yamada-Watanabe's theorem due to R ockner, Schmuland and Zhang.
The study undertaken in this thesis is in some sense pioneering since both classes
of stochastic partial di erential equations have not been the object of previous investigation,
to the best of our knowledge.
The results obtained are therefore original and constitute in our view signi cant
contribution to the nonlinear theory of stochastic parabolic equations.
