This paper studies the existence and uniqueness of local classical solutions to a class of degenerate semi-linear mixed problems. For degenerate parabolic equations, the standard solvability theory for parabolic equations does not apply. To address this issue, an approximation problem is constructed for the degenerate mixed problem, and solutions to the mixed problem are obtained via solutions to the approximation problem. First, the Schauder fixed point theorem is employed to obtain a local solution to the approximation problem. This solution is then extended to establish the existence of a global solution for the approximation problem. Subsequently, a comparison principle is established that simultaneously satisfies both the mixed problem and the approximating equation. Finally, by imposing certain appropriate constraints on the initial conditions of the mixed problem, the comparison principle, the internal Schauder estimate for the parabolic equation, and the Arzelà-Ascoli theorem are utilised to prove the existence of a unique local classical solution to the mixed problem.

In the report, the existence of the renormalized solutions of the Dirichlet boundary conditions, which can be satisfied by p(x)-Laplace elliptic equations with an optional low order term in bounded domains Ω, was studied. The low order term H was optional, and which satisfied certain growth conditions, the right end item f belonged to L¹ data. The function space theory, the truncation functions, and the gradient estimation methods were used to prove the existence of the renormalization solutions for p(x)-Laplace equations.