# On solvability of nonlocal boundary value problem for a differential equation of composite type

Keywords:
differential equation of composite type, nonlocal problem, regular solution, existence, uniqueness

### Abstract

We study the solvability in anisotropic Sobolev spaces of nonlocal in time problems for the differential equations of composite (Sobolev) type

$$u_{tt}+\left(\alpha\frac{\partial}{\partial t}+\beta\right)\Delta u+\gamma u=f(x,t),$$

$x = (x_1,\ldots , x_n) \in\Omega\subset R^n$, $t\in(0, T),$ $0 < T < +\infty$, $\alpha, \beta,$ and $\gamma$ are real numbers, and $f(x, t)$ is a given function. We prove theorems of existence and non-existence, uniqueness and non-uniqueness for regular solutions, those having all generalized Sobolev derivatives in the equation.

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Received

25-10-2021

How to Cite

*Mathematical notes of NEFU*, 28(4), pp. 90-100. doi: https://doi.org/10.25587/SVFU.2021.27.62.007.

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Section

Mathematics

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