Direct and inverse source problems for degenerate parabolic equations

Abstract

<jats:title>Abstract</jats:title><jats:p>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.</jats:p>

Reconstruction of time-dependent right-hand side in parabolic equations on disjoint domains

“…. There exists a large number of investigations considering identification of time dependent source, see e. g. [5,10,11,12,17,20,21], but results for interface problems of type (1)-( 6) are scare. In our previous work [14], we reconstruct the right-hand side in the time-fractional generalization of ( 1)-( 6) on disjoint domains, using decomposition technique.…”

“…So, this inverse problem belongs to the class of well-posed partial differential equations problems, see e.g. [10,11,12,17,21]. Despite of the special structure of the equation ( 14), it can be investigated for existence, uniqueness and continuous dependence from input data of the BVP (1)-( 6) and ( 7), following the methodology in [9].…”

“…For problems with unknown coefficients and right-hand side of equations, a number of results are obtained, see e.g. [10,11,12,13,17,20,21,22]. These are challenging and interesting inverse problems, which has many applications in various fields of physics and mechanics.…”

Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation

“…The literature related to inverse problems for degenerate parabolic PDEs, despite their fundamental importance and practical applications, is rather recent and scarce. For example, the inverse source problem was considered in Tort [38], Cannarsa et al [8], Cannarsa et al [7], Deng et al [10] and Hussein et al [18], where numerical reconstruction was also considered.…”

Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation

“…The literature related to inverse problems for degenerate parabolic PDEs, despite their fundamental importance and practical applications, is rather recent and scarce. For example, the inverse source problem was considered in Tort [39], Cannarsa, Tort, and Yamamoto [8], Cannarsa, Martinez and Vancostenoble [7], Deng and all [10] and Hussein and all [18], where numerical reconstruction was also considered. The inverse problem of recovering the first-order coefficient of degenerate parabolic equations is analyzed in Deng and Yang [11] and Kamynin [25].…”