Title: Coefficient Identification Problem with Integral Overdetermination Condition for Diffusion Equations Show Chinese title

Title: 带有积分过定条件的扩散方程系数识别问题

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. Show Chinese abstract

Abstract: 在本文中，我们研究一个非线性逆问题，旨在通过在空间域上的积分测量来恢复扩散方程 \( u_t - \Delta_x u - u_{yy} +a(t, x) u = f(t,x,y) \)中一个依赖于时间和部分空间变量的系数 $a(t, x)$。 此处 \(x \in G \subset \mathbb{R}^m\) 和 \(y \in (0, \pi)\)。 我们建立了局部和全局弱解的存在性和唯一性定理。 此外，我们证明了在问题数据足够光滑的情况下，逆问题存在唯一确定的强解（局部和全局）。 我们的方法结合了傅里叶方法和先验估计。 以前的研究已经处理了在整个空间上定义的抛物型方程类似的逆问题。