Sharp Lp-Lq estimates for evolution equations with damped oscillations

In this paper we derive sharp $L^p-L^q$ estimates, $1\leq p\leq q\leq \infty$ (including endpoint estimates as $L^1-L^1$ and $L^1-L^\infty$) for dissipative wave-type equations, under the assumption that the dissipation dampen the oscillations but it does not cancel them. We assume that the phase function $w$ is homogeneous of some degree $\sigma>0$ and that its Hessian matrix has maximal rank, including the critical case $\sigma=1$, while the dissipative term $a(\xi)>0$ may be inhomogeneous. The critical case includes waves with viscoelastic or structural damping, damped double dispersion equations and plate equations with rotational inertia, and so on. We also obtain the analogous results for fractional Schr\"odinger-type equations with a potential.