Evolution equations governed by the subdifferential of a convex composite function in finite dimensional spaces

Under quite general assumptions, we prove existence, uniqueness and regularity of a solution $U$ to the evolution equation $-U'(t)\in\partial(g\circ F)(U(t))$, $U(0)=u_0$, where $g:\mathbb{R}^q\rightarrow\mathbb{R}\cup\{+\infty\}$ is a closed proper convex function, $F:\mathbb{R}^p\rightarrow \mathbb{R}^q$ is a continuously differentiable mapping whose gradient is Lipschitz continuous on bounded subsets and $u_0\in\dom (g\circ F)$. We also study the asymptotic behavior of $U$ and give an application to nonlinear mathematical programming.