Linear inverse problems for a class of equations of Sobolev type

A. I. Kozhanov ; G. V. Namsaraeva

Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 2, pp. 153-171

Résumé

We study the solvability of inverse problems of finding together with the solution $u(x,t)$ also an unknown factor $q(t)$ in equation $$D^{2p}_t(u-\Delta u)+Bu=f_0(x,t)+q(t)h_0(x,t)$$ ($t\in (0,T)$, $x\in\Omega\subset \mathbb{R}^n$, $p$ is a natural number, $D^k_t=\frac{\partial^k}{\partial t^k}$, $\Delta$ is the Laplace operator with respect to the spatial variables, $B$ is a linear second-order differential operator, acting also on the spatial variables, $f_0(x,t)$ and $h_0(x,t)$ are given functions). Integral overdetermination condition is used as an additional condition in these problems. The existence and uniqueness theorems for regular solutions (i. e. having all the generalized derivatives in the sense of S.L. Sobolev, presenting in the equation) are proved.

Détail

Keywords: Sobolev type equation, inverse problem, unknown right-hand side, integral overdetermination, regular solution, solution existence, solution uniqueness.