Nonlinear damped partial differential equations and their uniform discretizations

TL;DR

This paper proves energy decay rates for nonlinear damped PDEs and develops discretization schemes that preserve these decay rates uniformly across discretization parameters, using convexity and observability techniques.

Contribution

It introduces discretization methods that maintain the continuous energy decay properties uniformly, applicable to a broad class of nonlinear PDEs including Schrödinger, wave, and plate equations.

Findings

Established sharp energy decay rates for nonlinear damped PDEs.

Designed discretization schemes that preserve decay rates uniformly.

Validated methods for various PDE models including nonlocal terms.

Abstract

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is inferred from the previous results. Our results cover, for instance, the Schr\"odinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, as well as certain classes of equations with nonlocal terms.