Convolution kernel identification problem for multi - dimensional time fractional diffusion - wave equation in bounded domain

We study the inverse problem on determining the convolution kernel of integral term in an initial boundary value problem for a multi - dimensional time fractional diffusion - wave equation with the uniformly elliptic operator in a divergent form. Moreover, as an overdetermination condition, a single observation at the point $x_0\in \Omega$ of the diffusion - wave process serves, where $\Omega\subset \mathbb{R}^n$ is a bounded domain. By the Fourier spectral method and fractional integro - differentiating technics the inverse problem is reduced to a convolution nonlinear Volterra integral equation of the second kind. The fixed point argument proves the local existence and global uniqueness results. Also the stability estimate for solution to inverse problem is obtained.

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