Asymptotic Behavior of Solutions to a Second-Order Gradient Equation of Pseudo-Convex Type

Consider in a real Hilbert space $H$ the second order gradient equation $$u''(t) = \nabla \phi(u(t)), \ \ \ t\geq0.$$ We state and prove several results on the weak or strong convergence of bounded solutions of this equation to minimizers of $\phi$, where $\phi\colon H\to \mathbb{R}$ is a continuously differentiable, pseudo-convex function with ${\rm Argmin}\,\phi\neq\varnothing$. Our results extend previous results in the literature that are related to the case when $\phi$ is convex.