Linear inverse problems of spatial type for quasiparabolic equations
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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