CONVERGENCE TO EQUILIBRIUM FOR A TIME SEMI-DISCRETE DAMPED WAVE EQUATION
Morgan Pierre, Laboratoire de Mathématiques et Applications UMR CNRS 7348, Université de Poitiers, Téléport 2-BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France, Philippe Rogeon · Journal of Applied Analysis & Computation · 2016
We prove that the solution of the backward Euler scheme applied to a damped wave equation with analytic nonlinearity converges to a stationary solution as time goes to infinity. The proof is based on the Łojasiewciz-Simon inequality. It is much simpler than in the continuous case, thanks to the dissipativity of the scheme. The framework includes the modified Allen-Cahn equation and the sine-Gordon equation.