1. CONVERGENCE IN RELATIVE ERROR FOR THE POROUS MEDIUM EQUATION IN A TUBE
- Author
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Audrito, Alessandro, Gárriz, Alejandro, Quirós, Fernando, Department of Mathematics - ETH, Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Departamento de Matematicas [Madrid], Universidad Autónoma de Madrid (UAM), European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska–Curie grant agreement 892017 (LNLFB-Problems), ANR project DEEV ANR-20-CE40-0011-01, MCIN/AEI (Spain) through projects MTM2017-87596-P, PID2020-116949GB-I00, RED2018-102650-T and the ICMAT-Severo Ochoa grant CEX2019-000904-S, and ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020)
- Subjects
Mathematics - Analysis of PDEs ,Traveling waves ,FOS: Mathematics ,Long-time behaviour ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Porous medium diffusion in tubes ,Convergence in relative error ,2010 Mathematics Subject Classification. 35K57, 35K65, 35C07, 35K55 ,Analysis of PDEs (math.AP) - Abstract
Given a bounded domain $D \subset \mathbb{R}^N$ and $m > 1$, we study the long-time behaviour of solutions to the Porous Medium equation (PME) posed in a tube \[ \partial_tu = Δu^m \quad \text{ in } D \times \mathbb{R}, \quad t > 0, \] with homogeneous Dirichlet boundary conditions on the boundary $\partial D \times \mathbb{R}$ and suitable initial datum at $t=0$. In two previous works, Vázquez and Gilding & Goncerzewicz proved that a wide class of solutions exhibit a traveling wave behaviour, when computed at a logarithmic time-scale and suitably renormalized. In this paper, we show that, for large times, solutions converge in relative error to the Friendly Giant, i.e., the unique nonnegative solution to the PME posed in the section $D$ of the tube (with homogeneous Dirichlet boundary conditions) having a special self-similar form. In addition, sharp rates of convergence and uniform bounds for the location of the free boundary of solutions are given.
- Published
- 2022