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 $u_t-{\mathrm{\Delta }}_{x}u-u_{yy}+a(t,x)u=f(t,x,y)$, 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.