Parabolic Problems with Strong Degeneracy at the Spatial Boundary
Jerome A. Goldstein, Chin-Yuan Lin · 2023
This chapter derives parabolic partial differential equation, and explains the case when the diffusion coefficient vanishes on the spatial boundary. The researchers&s; approach is based on the Crandall-Liggett-Benilan theorem, the cornerstone of the nonlinear semigroup approach. In order to use this theorem, users make the restrictive hypothesis that depend only on x and u , not on u . The absence of u dependence leads to a dissipative operator. Under certain circumstances, u dependence leads to a locally quasi-dissipative operator, and recently Dor roll and Rieder, in a deep and very interesting paper, established local (in time) well-posedness for the mixed initial-boundary value problem with u dependence in the diffusion coefficient.