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Exact traveling-wave solutions of the nonlinear Klein-Gordon equation with dissipation are found. The solutions are discussed in the context of propagating nonlinear mechanical waves as well as domain walls (solutions interpolating between two local minima in the potential) and relaxation modes (solutions interpolating between a local minimum and a local maximum in the potential) in systems undergoing second-order phase transitions. Using the Lyapunov method the stability of traveling waves is analyzed.Explicit solutions for domain walls are found and shown to be asymptotically stable. While domain walls propagate with a velocity uniquely determined by the parameters of the model, relaxation modes can move with arbitrary velocities. Among those solutions only one solution propagating with a characteristic velocity u, is asymptotically stable. Other solutions are demonstrated to be Lyapunov unstable.It is found that relative values of the mass density and the damping coeScient are crucial for the dynamic critical behavior, which changes from U, -(T, -T)' for small densities to v. that is temperature independent for large densities.