Lp-Lq decay estimates for Evolution Operators

Abstract

In this project, we are interested in Lp-Lq decay estimates (not necessarily on the conjugate line) in time for linear hyperbolicequations or more in general, p-evolution equations. The results are derived by developing a suitable WKB analysis.We plan to apply these estimates to study semi-linear problems. In particular, we are interested in proving results about global existence (in time) of the solution, possibly assuming small initial data. Here we plan to understand in which cases the decay rates of solutions to the semi-linear problems coincide with those ones for the corresponding linear problem, and in which other cases a loss of decay appears. Then the question for the exact loss of decay appears.So methods to show optimality should be developed.We plan to study both models with constant coefficients and with time-dependent coefficients as well. In the case of time-dependent coefficients, we will assume suitable regularity and a sufficient control of the oscillations. Also, the interaction of the time-dependent coefficients will be studied to avoid bad influence on the asymptotic profile, or to obtain better decay estimates.In a first moment, we will mainly consider wave-type equations, possibly with damping terms, and with nonlocal terms, like fractional powers of the Laplacian. In this way we cover external and structural damping up to the visco-elastic case. Finally, we plan to study higher order equations and, if possible, first-order systems, $p$-evolution equations and problems in an abstract setting. (AU)