Well-Posedness of One-Dimensional Nonlinear Diffusion Equations with a Fourth-Type Dynamic Boundary Condition

Abstract: This paper addresses the local well-posedness of the Cauchy problem for a one-dimensional diffusion equation equipped with a dynamic boundary condition and an additional boundary condition that renders the one-dimensional Laplace operator self-adjoint. The equation serves as a model for describing filtration in aquaria, originally introduced by the author and Kitahata. The boundary condition treated in this work differs from classical types such as Dirichlet, Neumann, and Robin conditions; we refer to it as the fourth or FK-type boundary condition. The boundary condition is designed to capture interactions between the two boundaries in the context of filtration. The framework for establishing well-posedness is based on L^p-L^q maximal regularity classes.