On the Non-local Problem for a Boussinesq Type Equations

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.

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