Decay Estimates for a Lamé Inverse Problem Involving Source and Damping Term with Variable-Exponent Nonlinearities

We investigate an inverse problem involving source and damping term with variable-exponent nonlinearities. We establish adequate conditions on the initial data for the decay of solutions as the integral overdetermination approaches zero over time within an acceptable range of variable exponents. This class of inverse problems, where internal terms such as source and damping are to be determined from indirect measurements, has significant relevance in real-world applications—ranging from geophysical prospecting to biomedical engineering and materials science. The accurate identification of these internal mechanisms plays a crucial role in optimizing system performance, improving diagnostic accuracy, and constructing predictive models. Therefore, the results obtained in this study not only contribute to the theoretical understanding of nonlinear dynamic systems but also provide practical insights for reconstructive analysis and control in applied settings. The asymptotic behavior and decay conditions we derive are expected to be of particular interest to researchers dealing with stability, uniqueness, and identifiability in inverse problems governed by nonstandard growth conditions.