ON FAST AND SLOW TIMES IN MODELS WITH DIFFUSION

The linear Kelvin-Voigt operator L is a typical example of wave operator L0 perturbed by higher - order viscous terms . If P is a prefixed boundary - value problem for L, when small parameter goes to zero L turns into L 0 and P into a problem P0 with the same initial - boundary conditions of P. Boundary - layers are missing and the related control terms depending on the fast time are neglegible. In a small time interval, the wave behavior is a realistic approximation of u when small parameter tends to zero. On the contrary, when t is large, diffusion effects should prevail and the behavior of u when small parameter tends to zero and time tends to infinity should be analyzed. For this, a suitable functional corrispondence between the Green functions G and G0 of P and P0 is achieved and its asymptotic behavior is rigorously examined. As consequence, the interaction between diffusion effects and pure waves is evaluated by means of the slow time. The time - intervals in which pure waves are propagated nearly indisturbed or damped oscillations predominate, are determined.