Direct and inverse source problems for degenerate parabolic equations

  • 1 Department of Applied Mathematics, University of Leeds, LS2 9JT, Leeds, United Kingdom
  • 2 Department of Mathematics, University of Baghdad, College of Science, Baghdad, Iraq
  • 3 National Research Nuclear University MEPhI, Kasirskoe shosse, 31, 115409, Moscow, Russia
M. S. Hussein, Daniel Lesnic
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  • Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
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, Vitaly L. KamyninORCID iD: and Andrey B. Kostin


Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.