Stability and regularity analysis of nonlinear wave equations with localised internal and Ventcell boundary conditions

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This paper primarily focuses on analysing the exponential decay stability and regularity of the solution to a nonlinear wave equation with localised internal damping, characterised by a Carathéodory function and Ventcel-Dirichlet boundary conditions. Additionally, the regularity of the solution within this framework is investigated. Stability through exponential decay is established by formulating fresh Lyapunov functions and employing multiplier techniques. It's essential to underline the recognition of the memory term's influence at the boundary on the solution, thereby introducing a degree of uncertainty into the analysis.