Controllability of a System with Nonlinear Damping Devices and Nonlinear Source Terms in Elasticity Problems: Existence, Time Blow-up, and Numerical Results

Swelling soil problems arise in various real-world applications, such as geomechanics, biomedical engineering, and hydrogel-based materials, where fluid interaction with elastic structures influences mechanical stability. In this study, we investigate a swelling soil system incorporating two nonlinear variable exponent damping and source terms, which provide a more adaptable framework for capturing heterogeneous material behaviors and evolving energy dissipation mechanisms. Using the Faedo-Galerkin method and the Banach Contraction Theorem, we establish the local existence and uniqueness of weak solutions under suitable conditions on the variable exponent functions. Furthermore, we demonstrate the global existence of solutions and identify conditions leading to finite-time blow-up, offering insights into stability and failure prediction in porous-elastic media. To validate our theoretical findings, we present numerical simulations illustrating the blow-up behavior, emphasizing the role of variable exponent damping in influencing system dynamics.