Duke Applied Mathematics

Publications [#370567] of Thomas P. Witelski

Papers Published

1. Bowen, M; King, JR; Witelski, TP, CAUCHY-DIRICHLET PROBLEMS FOR THE POROUS MEDIUM EQUATION, Discrete and Continuous Dynamical Systems Series A, vol. 43 no. 3-4 (March, 2023), pp. 1143-1174, American Institute of Mathematical Sciences (AIMS) [doi]
   (last updated on 2026/01/19)

   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.