Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method

In this work we study the asymptotic behavior of a class of damped second-order gradient systems $$\ddot{u}(t) + a\dot{u}(t) + \nabla W(u(t)) = 0,$$ under assumptions ensuring local convexity of the potential near equilibrium and coercivity at infinity. By introducing a Lyapunov functional adapted to the geometry of the system, we establish uniform asymptotic stability of the equilibrium for all $a \in (0,a_0]$, together with exponential decay when the potential satisfies a quadratic control near its minimum. Furthermore, complementary arguments based on semigroup theory reveal the existence of a global attractor. We also present numerical simulations for some $W$ potentials that illustrate the behavior of trajectories near equilibrium, in both dissipative and conservative regimes.