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Volume 6, Number 4, 2016, Pages 1041-1048                                                                DOI:10.11948/2016067
Convergence to equilibrium for a time semi-discrete damped wave equation
Morgan Pierre,Philippe Rogeon
Keywords:{\L}ojasiewicz-Simon inequality, gradient-like equation, backward Euler scheme.
      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 {\L}ojasiewciz-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.
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