Abstract.

In this paper we prove uniqueness in the inverse boundary value problem for the three coefficient functions in the porous medium equation with an absorption term \(\epsilon \partial_t u-\nabla \cdot (\gamma \nabla u^m)+\lambda u^q=0\) with \(m\gt 1\), \(m^{-1}\lt q\lt \sqrt{m}\), with the space dimension 2 or higher. This is a degenerate parabolic type quasilinear PDE which has been used as a model for phenomena in fields such as gas flow (through a porous medium), plasma physics, and population dynamics. In the case when \(\gamma=1\) a priori, we prove unique identifiability with data supported in an arbitrarily small part of the boundary. Even for the global problem we improve on previous work by obtaining uniqueness with a finite (rather than infinite) time of observation and also by introducing the additional absorption term \(\lambda u^q\).

Keywords

  1. inverse problems
  2. porous medium equation
  3. nonlinear parabolic equations

MSC codes

  1. 35R30
  2. 35K55
  3. 35K59
  4. 35K65

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 162 - 185
ISSN (online): 1095-7154

History

Submitted: 15 December 2021
Accepted: 6 September 2022
Published online: 28 January 2023

Keywords

  1. inverse problems
  2. porous medium equation
  3. nonlinear parabolic equations

MSC codes

  1. 35R30
  2. 35K55
  3. 35K59
  4. 35K65

Authors

Affiliations

School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China.
Department of Mathematics, Universität Bielefeld, 33615 Bielefeld, Germany.
Gunther Uhlmann
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA. Institute for Advanced Study of the Hong Kong University of Science and Technology, Hong Kong.

Funding Information

Funding: The work of the first author was supported by the National Natural Science Foundation of China grants 11931011, 11971333. The work of the second author was supported by the Collaborative Research Center, membership 1283, Universitat Bielefeld. The work of the third author was partially supported by the National Science Foundation, a Walker Family Professorship at the University of Washington, a Si Yuan Professorship at IAS, HKUST, and a Simons Foundation Fellowship. Part of this work was done while the third author was visiting IPAM in Fall 2021.

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