We study the existence and asymptotic convergence when t --> +infinity for the trajectories generated bydel(2) f (u(t), epsilon(t))(u) over dot (t) + (epsilon) over dot (t) partial derivative(2)f/partial derivative epsilon partial derivative x (u(t)) + del f(u(t), epsilon(t)) = 0,where {f(., epsilon)}(epsilon>0) is a parametric family of convex functions which approximates a given convex function f we want to minimize, and epsilon(t) is a parametrization such that epsilon(t) -->, 0 when t --> +infinity. This method is obtained from the following variational characterization of Newton's method:(P-t(epsilon))( )u(t) is an element of Argmin {f (x, epsilon(t)) - e(-t )[del f (u(0), epsilon(0)),( )x] : x is an element of H}where H is a real Hilbert space, We find conditions on the approximating family f (., epsilon) and the parametrization epsilon(t) to ensure the norm convergence of the solution trajectories u(t) toward a particular minimizer of f. The asymptotic estimates obtained allow us to study the rate of convergence as well. The results are illustrated through some applications to barrier and penalty methods for linear programming, and to viscosity methods for an abstract noncoercive variational problem. Comparisons with the steepest descent method are also provided.