Morgan Pierre (Laboratoire de Mathématiques et Applications)
A celebrated result of S. Lojasiewicz states that every bounded solution of a gradient flow associated to an analytic function converges to a steady state as time goes to infinity. Convergence rates can also be obtained. These convergence results have been generalized to a large variety of finite or infinite dimensional gradient-like flows. The fundamental example in infinite dimension is the semilinear heat equation with an analytic nonlinearity. In this talk, we show how some of these results can be adapted to time discretizations of gradient-like flows, in view of applications to PDEs such as the Allen-Cahn equation, the sine-Gordon equation, the Cahn-Hilliard equation, the Swift-Hohenberg equation, or the phase-field crystal equation.