Abstract
This article addresses an ill-posed problem related to the biharmonic equation within a hemispherical domain. It explores the absence of continuous dependency of solutions on the problem's input data and establishes a framework for conditional well-posedness. The study includes a theorem characterizing the stability conditions for the biharmonic equation, utilizing Fourier series and regularization techniques for approximating solutions. A metric-based estimation of errors between exact and approximate solutions is also provided. The article contributes to the understanding and resolution of ill-posed problems in mathematical physics, emphasizing the selection of optimal regularization parameters for improved solution stability and accuracy.