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We establish the decay of the solutions of the damped wave equations in onedimensional space for the Dirichlet, Neumann, and dynamic boundary conditionswhere the damping coefficient is a function of space and time. The analysis isbased on the study of the corresponding hyperbolic systems associated with theRiemann invariants. The key ingredient in the study of these systems is the useof the internal dissipation energy to estimate the difference of solutions withtheir mean values in an average sense.

Exponential decay of solutions of damped wave equations in one dimensional space in the $L^p$ framework for various boundary conditions