Linear inverse problems of spatial type for quasiparabolic equations

E. V. Akimova   ; A. I. Kozhanov

Matematičeskie zametki SVFU, Tome 25 (2018) no. 3, pp. 3-17

Résumé

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 $\mathbb{R}^n,$ $t\in(0,T),$ $0<T<+\infty,$ $f(x,t)$ 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).

Détail

Keywords: linear inverse problem, boundary overdetermination condition, regular solutions, uniqueness.
Mots-clés : quasiparabolic equations, existence