Well-Posedness of the Cauchy Problem for One-Dimensional Nonlinear Diffusion Equations with Dynamic and Fourth-Type Boundary Conditions in the Lp Lq Maximal Regularity Setting

TL;DR

This paper proves the local well-posedness of a one-dimensional nonlinear diffusion equation with a novel boundary condition, using L^p-L^q maximal regularity, to model filtration processes with boundary interactions.

Contribution

It introduces and analyzes a new FK-type boundary condition for diffusion equations, establishing well-posedness in the L^p-L^q maximal regularity framework.

Findings

Established local well-posedness for the model equation.

Demonstrated the applicability of maximal regularity techniques.

Provided a mathematical foundation for filtration models with complex boundary interactions.

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.