Cauchy problem for singular-degenerate porous medium type equations: well-posedness and Sobolev regularity

TL;DR

This paper investigates the well-posedness and Sobolev regularity of solutions to reaction-diffusion equations with porous medium type degeneracy and singularities, motivated by biofilm growth models.

Contribution

It introduces new analytical results on existence, uniqueness, and regularity for a class of degenerate and singular quasilinear PDEs using advanced mathematical techniques.

Findings

Proved well-posedness of solutions under certain conditions.

Established Sobolev regularity results for solutions.

Applied m-accretive operator theory and Fourier analysis techniques.

Abstract

Motivated by models for biofilm growth, we consider Cauchy problems for quasilinear reaction diffusion equations where the diffusion coefficient has a porous medium type degeneracy as well as a singularity. We prove results on the well-posedness and Sobolev regularity of solutions. The proofs are based on m-accretive operator theory, kinetic formulations and Fourier analytic techniques.