In this paper, we investigate the inverse problem of determining an unknown time-dependent source term in a semilinear pseudo-parabolic equation with variable coefficients and a Dirichlet boundary condition. The unknown source term is recovered from additional measurement data expressed as a weighted spatial average of the solution. By employing Rothe's time-discretisation method, we prove the existence and uniqueness of a weak solution under a smallness condition on the problem data. We also present a numerical scheme for computations.

翻译：本文研究了一类具有变系数和Dirichlet边界条件的半线性拟抛物方程中未知时间依赖源项的反演问题。该未知源项通过表示为解的空间加权平均的附加测量数据得以重构。通过采用Rothe时间离散化方法，我们在问题数据满足小性条件下证明了弱解的存在唯一性。同时，本文还提出了一种用于数值计算的求解格式。