Linear inverse problems of spatial type for quasiparabolic equations

  • Akimova Ekaterina V., ek.v.akimova@gmail.com Novosibirsk State University, Pirogova Street, 1, Novosibirsk 630090, Russia
  • Kozhanov Aleksandr I., kozhanov@math.nsc.ru Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia
Keywords: linear inverse problem, quasiparabolic equations, boundary overdetermination condition, regular solutions, existence, uniqueness

Abstract

We study solvability of the inverse problems for finding both the solution $u(x, t)$ and the coefficient $q(x)$ in the equation

$$(−1)^{m+1}\frac{\partial^{2m+1}u}{\partial t^{2m+1}}+\Delta u + \mu u=f(x,t)+q(x)h(x,t),$$

where $x=(x_1,\dots , x^n)\in\Omega,$ $\Omega$ is a bounded domain in $R^n,$ $t\in(0, T),$  $0 < T <+\infty,$ $f(x, t)$ and $h(x, t)$ are given functions, $\mu$ is a given real, $m$ is a given natural, and $\Delta$ is necessary due to presence of the additional unknown function $q(x)$), the boundary overdetermination condition is used in the article (with $t = 0$ or $t = T$).
For the problems under study, the existence and uniqueness theorems for regular solutions are proved (all derivatives are the Sobolev generalized derivatives).

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How to Cite
Akimova, E. and Kozhanov, A. (&nbsp;) “Linear inverse problems of spatial type for quasiparabolic equations”, Mathematical notes of NEFU, 25(3), pp. 3-17. doi: https://doi.org/10.25587/SVFU.2018.99.16947.
Section
Mathematics