AN INVERSE PROBLEM FOR THE POROUS MEDIUM EQUATION WITH PARTIAL DATA AND A POSSIBLY SINGULAR ABSORPTION TERM

Cătălin I. Cărstea, Tuhin Ghosh, Gunther Uhlmann

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 є∂tu - ∇ ∙ (γ∇um) + λuq = 0 with m > 1, m -1 < q < √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 γ = 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 λuq.

Original languageEnglish
Pages (from-to)162-185
Number of pages24
JournalSIAM Journal on Mathematical Analysis
Volume55
Issue number1
DOIs
StatePublished - Feb 2023

Keywords

  • inverse problems
  • nonlinear parabolic equations
  • porous medium equation

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