2020, Kamynin, V. L., Kostin, A. B., Камынин, Виталий Леонидович, Костин, Андрей Борисович

© 2019, Springer Science+Business Media, LLC, part of Springer Nature.We consider the inverse multifactor source problem for a uniformly parabolic equation with integral overdetermination conditions and study the existence, uniqueness, and stability of a solution. We obtain two forms of sufficient conditions for the unique solvability and present examples of inverse problems satisfying the assumptions of the proved theorems.

2020, Hussein, M. S., Lesnic, D., Kamynin, V. L., Kostin, A. B., Камынин, Виталий Леонидович, Костин, Андрей Борисович

© 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.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- A nd 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.

2021, Kamynin, V. L., Kostin, A. B., Камынин, Виталий Леонидович, Костин, Андрей Борисович

We consider the inverse problem of finding the coefficient before u(t, x) in a higher-order parabolic equation, which is not assumed to be uniformly parabolic and can admit weak degeneracy. The required coefficient is considered to depend only on the spatial variable x is an element of [0, l]. Additional information is taken in the form of an integral over the variable t is an element of [0, T] of the solution with a given weight function (integral observation). The initial condition and m boundary conditions of the first kind (2m order of the equation) are specified in a standard way. It is assumed that the leading coefficient rho before u(t) in the equation is non-negative, and its reciprocal 1/rho belongs to the space L-q(Q) for some q > 1. For the considered inverse problem, existence and uniqueness theorems for the generalized solution are proved. In the course of its research, the corresponding theorems on the solvability of the direct problem were formulated and proved. In this case, the approaches and results of the well-known work of S. N. Kruzhkov(1979) were used. In the conclusion, we give an example of the inverse problem for which the conditions of the theorems proved are satisfied. It is shown that for all sufficiently large values of T > 0 its solution exists, is unique, and an estimate for the required coefficient is written out.

2019, Kamynin, V. L., Kostin, A. B., Камынин, Виталий Леонидович, Костин, Андрей Борисович

© 2019 Published under licence by IOP Publishing Ltd.We find two type conditions sufficient for unique solvability of inverse problem of source determination in degenerate parabolic equation with two independent variables.