ON FAST AND SLOW TIMES IN MODELS WITH DIFFUSION

Abstract

The linear Kelvin–Voigt operator ℒε is a typical example of wave operator ℒ0 perturbed by higher-order viscous terms as εuxxt. If [Formula: see text] is a prefixed boundary value problem for ℒε, when ε = 0, ℒε turns into ℒ0 and [Formula: see text] into a problem [Formula: see text] with the same initial–boundary conditions of [Formula: see text]. Boundary layers are missing and the related control terms depending on the fast time are negligible. In a same time interval, the wave behavior is a realistic approximation of uε when ε → 0. On the contrary, when t is large, diffusion effects should prevail and the behavior of uε for ε → 0 and t → ∞ should be analyzed. For this, a suitable functional correspondence between the Green functions [Formula: see text] and [Formula: see text] of [Formula: see text] and [Formula: see text] is derived and its asymptotic behavior is rigorously examined. As a consequence, the interaction between diffusion effects and pure waves is evaluated by means of the slow time εt; the main results show that in time intervals as (ε, 1/ε) pure waves are quasi-undamped, while damped oscillations predominate as from the instant t > 1/ε.