Inverse source problems for the polyharmonic heat equation
Ismailov M.I. Suragan D.
2026Walter de Gruyter GmbH
Journal of Inverse and Ill-Posed Problems
2026
In the present paper, we consider the inverse problem for the polyharmonic heat equation, aiming to recover a space/time heat source F along with the temperature distribution u(x, t). The problem is governed by the higher-order heat equation (∂t + (−Δ)l)u = F in a finite cylindrical domain Ω × (0, T] within the half-space ℝd × [0, +∞), where l, d ≥ 1. This is done through utilizing the values of u and its normal derivatives up to order l − 1 on a given lateral surface of the cylinder, as well as the initial value and time-averaged data (or time observation measurement) for space-dependent source and space-averaged data (energy/mass measurement) for time-dependent source. We determine that the integer k = [ 4dl] + 2 is admissible for the required degree of data regularity. The well-posedness of the classical solution to the inverse problem are established by employing the method of series expansion in terms of eigenfunctions for the Dirichlet poly-Laplacian. Weyl-type eigenvalue inequalities play a key role in our derivations.
Fourier method , Polyharmonic heat equation , space-dependent heat source , time-dependent heat source , Weyl’s asymptotic formula
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Department of Mathematics, Gebze Technical University, Gebze, Kocaeli, 41400, Turkey
Center for Mathematics and its Applications, Khazar University, Baku, AZ1096, Azerbaijan
Department of Mathematics, Nazarbayev University, Astana, 010000, Kazakhstan
Department of Mathematics
Center for Mathematics and its Applications
Department of Mathematics
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