A boundary value problem for a class of anisotropic stochastic degenerate parabolic-hyperbolic equations

We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain O = O x O where a Neumann boundary condition is imposed on partial differential O, x O, the hyperbolic boundary, and a Dirichlet condition is imposed on O x partial differential O, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem. & COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

In this study, we establish the well-posedness of an initial-boundary value problem for a stochastic nonlinear parabolic-hyperbolic equation of mixed type. The problem is defined on a space domain O = O x O, where a Neumann boundary condition is imposed on the hyperbolic boundary O x O and a Dirichlet condition is imposed on the parabolic boundary O x O. Our analysis highlights the new strong trace theorem for this class of equations, which is crucial for the uniqueness of the kinetic solution, and the new averaging lemma, which is an important part of proving the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems and prove the convergence of their solutions to the solution of the initial-boundary value problem.