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Cauchy-Dirichlet problems for the porous medium equation

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

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Mark Bowen

Professor of Theoretical Mechanics

Thomas P Witelski


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.


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.

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
Keywords Porous medium equation, similarity solutions, conserved quantities, nonlinear diffusion, Dirichlet problem
Public URL
Publisher URL
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


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