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We establish sharp energy decay rates for a large class of nonlinearlyfirst-order damped systems, and we design discretization schemes that inheritof the same energy decay rates, uniformly with respect to the space and/or timediscretization parameters, by adding appropriate numerical viscosity terms. Ourmain arguments use the optimal-weight convexity method and uniformobservability inequalities with respect to the discretization parameters. Weestablish our results, first in the continuous setting, then for spacesemi-discrete models, and then for time semi-discrete models. The fulldiscretization is inferred from the previous results. Our results cover, forinstance, the Schr\"odinger equation with nonlinear damping, the nonlinear waveequation, the nonlinear plate equation, as well as certain classes of equationswith nonlocal terms.

Nonlinear damped partial differential equations and their uniform discretizations