Abstract

The problem of finding a solution, satisfying the non-local condition \(u(\xi_{0})=\alpha u(+0)+\varphi\) in time for the Boussinesq type equation of the form \(u_{tt}+Au_{tt}+Au=f\) is studied in the article. Here \(\alpha\) and \(\xi_{0}\), \(\xi_{0}\in(0,T],\) are the given numbers, \(A:H\rightarrow H\) is the self-adjoint, unbounded, positive operator defined in the Hilbert separable space \(H\). By using the Fourier method, it was shown that the solution to the problem exists and is unique. The effect of parameter \(\alpha\) on the existence and uniqueness of the solution is studied in the article. The inverse problem of determining the right-hand side of the equation is also considered.

中文翻译：

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关于Boussinesq型方程的非局部问题

 抽象的

寻找解的问题，对于形式为 \(u_{文中研究了tt}+Au_{tt}+Au=f\)。这里 \(\alpha\) 和 \(\xi_{0}\), \(\xi_{0}\in(0,T],\) 是给定的数字，\(A:H\rightarrow H\)是希尔伯特可分空间\(H\)中定义的自伴、无界、正算子。通过使用傅立叶方法，证明了该问题的解存在且唯一。参数\(\alpha)的作用。 \) 文中还研究了解的存在唯一性，并考虑了确定方程右边的反问题。