Coefficient Identification Problem with Integral Overdetermination Condition for Diffusion Equations

TL;DR

This paper addresses a nonlinear inverse problem for diffusion equations with an integral overdetermination condition, establishing existence, uniqueness, and regularity of solutions using Fourier methods and a priori estimates.

Contribution

It introduces new theorems on existence and uniqueness of solutions for a coefficient identification problem with integral overdetermination in diffusion equations.

Findings

Proved existence of local and global weak solutions.

Established uniqueness of solutions under certain conditions.

Demonstrated existence of strong solutions with sufficient data smoothness.

Abstract

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.