New Lp-Lq estimates for the wave equation with viscoelastic dissipation

In this talk, we show recent Lp-Lq estimates, for general (p,q), obtained in collaboration with M.R. Ebert, for the solution to the Cauchy problem for the visco-elastic damped wave equation. These results complete the picture that was started by Y. Shibata (MMAS, 2000), who obtained the endpoint estimates (p,q)=(1,1) and (p,q)=(1,∞). We show that optimal Lp-Lq estimates in the general case cannot be derived by a simple interpolation due to the “noneffective” nature of the damping term. We show the analogous but different result which holds for other strongly damped p-evolution models, as the plate equation. We mention a straightforward application of the obtained estimates to derive global existence of small data solutions for problems with critical or supercritical power nonlinearities.