Coefficient Identification Problem with Integral Overdetermination Condition for Diffusion Equations

In this paper, we investigate a nonlinear inverse problem aimed at recovering a coefficient $a(t, x)$, dependent on both time and a subset of spatial variables, in a diffusion equation $u_t - \Delta_x u - u_{yy} +a(t, x) u = f(t,x,y)$, using an additional measurement given as an integral over the spatial domain. Here $x \in G \subset \mathbb{R}^m$ and $y \in (0, \pi)$. We establish theorems on the existence and uniqueness of both local and global weak solutions. Furthermore, we demonstrate that, under sufficient smoothness of the problem data, there exists a uniquely determined strong solution (both local and global) to the inverse problem. Our approach combines the Fourier method with a priori estimates. Previous studies have addressed similar inverse problems for parabolic equations defined over the entire space.