Calculus of Variations and Geometric Measure Theory

G. Bellettini - C. Geldhauser - M. Novaga

Convergence of a semidiscrete scheme for a forward-backward parabolic equation

created by novaga on 26 Jun 2012
modified on 15 May 2016


Published Paper

Inserted: 26 jun 2012
Last Updated: 15 may 2016

Journal: Advances Differential Equations
Volume: 18
Number: 5/6
Pages: 495-522
Year: 2013


We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation $u_t= (W'(u_x))_x$ with periodic boundary conditions in one space dimension, where $W$ is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. Using an approximation argument, we show that it is possible to flow initial data with derivative in the concave region $\{W''<0\}$ of $W$, where the backward character of the equation manifests. It turns out that the limit equation depends on the way we approximate the initial data in the unstable region.