Grant number: 21/01743-3 Support type: Scholarships in Brazil - Post-Doctorate Effective date (Start): September 01, 2021 Effective date (End): August 31, 2023 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis Principal researcher: Marcelo Rempel Ebert Grantee: Halit Sevki Aslan Home Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil Abstract In this project, we are interested in the asymptotic behavior (in time) of solutions for some linear and semi-linear hyperbolicequations or more in general, for evolution equations. The results are derived by developing a suitable Fourier Analysis in the phase space and by applying the well known stationary phase method.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 as~$t\to\infty$. Also, we plan to apply the obtained linear estimates to prove global existence of small data solutions to associate semilinear problems with power nonlinearities. Also, in the subcritical case, to prove non-existence and blow-up results. (AU)