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 $\varphi$ which is nonconvex in the gradient $u_x$. Typical examples of integrands are $\varphi (u_x)=\log (1+u_{x}2)$ and $\varphi (u_x)=(1-u_{x}2)2$, 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).