**Abstract**

We plan to study decay estimates for linear hyperbolic equations. We are interested in different types of decay estimates in time, in Sobolev spaces. On the one hand, we will study the setting. On the other hand, we will study estimates, not necessarily on the conjugate line, distinguishing between low-frequencies and high-frequencies estimates. We plan to apply some of the derived estimates to study semilinear problems. In particular, we are interested in proving results of global existence of the solution, possibly assuming small initial data. In the case of small data solution, we will study in which cases the decay rates of the semilinear problems are the same of the linear one, and in which other cases a loss of decay appears. We plan to study both models with constant coefficients and with time-dependent coefficients. In the case of time-dependent coefficients, we will assume suitable regularity and a sufficient control on the oscillations, to guarantee the desired result. 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. We also plan to study higher order equations and possibly first-order systems, evolution equations and problems in abstract setting. (AU)

Scientific publications
(12)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)