Some results on gradient flows of nonconvex functionals in one dimension

We are concerned with the problem of defining a reasonable notion of global solution to the partial differential equation obtained as the $L2$-gradient flow of an integral functional with an integrand $\phi$ which is nonconvex in the gradient $u_x$. Typical examples of integrands are $\phi(u_x) = \log(1+u_x2)$ and $\phi(u_x) = (1-u_x2)2$, which give raise to backward-forward parabolic equations. We discuss some results related to the approximation of such equations; we construct also a solution for a particularly simple nonconvex integrand. These results are part of a series of papers (mainly in progress) in collaboration with G. Fusco (Univ. l'Aquila), N. Guglielmi (Univ. l'Aquila), M. Novaga (Univ. Pisa), E. Paolini (Univ. Firenze).