Cauchy-Dirichlet problems for the porous medium equation

Bowen, Mark; King, John R.; Witelski, Thomas P

Abstract

We consider the porous medium equation subject to zero-Dirichlet conditions on a variety of two-dimensional domains, namely strips, slender domains and sectors, allowing us to capture a number of different classes of behaviours. Our focus is on intermediate-asymptotic descriptions, derived by formal arguments and validated against numerical computations. While our emphasis is on non-negative solutions to the slow-diffusion case, we also derive a number of results for sign-change solutions and for fast diffusion. Self-similar solutions of various kinds play a central role, alongside the identification of suitable conserved quantities. The characterisation of domains exhibiting infinite-time hole closure is a particular upshot and we highlight a number of open problems.

Citation

Bowen, M., King, J. R., & Witelski, T. P. (2023). Cauchy-Dirichlet problems for the porous medium equation. Discrete and Continuous Dynamical Systems - Series A, 43(3&4), 1143-1174. https://doi.org/10.3934/dcds.2022182

• Journal Article Type: Article
• Acceptance Date: Dec 6, 2022
• Online Publication Date: Dec 1, 2022
• Publication Date: Mar 1, 2023
• Deposit Date: Feb 22, 2023
• Publicly Available Date: Feb 22, 2023
• Journal: Discrete and Continuous Dynamical Systems- Series A
• Print ISSN: 1078-0947
• Electronic ISSN: 1553-5231
• Publisher: American Institute of Mathematical Sciences (AIMS)
• Peer Reviewed: Peer Reviewed
• Volume: 43
• Issue: 3&4
• Pages: 1143-1174
• DOI: https://doi.org/10.3934/dcds.2022182
• Keywords: Porous medium equation, similarity solutions, conserved quantities, nonlinear diffusion, Dirichlet problem
• Public URL: https://nottingham-repository.worktribe.com/output/17657829
• Publisher URL: https://www.aimsciences.org/article/doi/10.3934/dcds.2022182
• Additional Information: Received: March 2022, Revised: November 2022, Early access: December 2022, Published: March 2023Dedicated to Professor Juan Luis Vázquez on the occasion of his 75th birthday