Oscillatory integrals for Evolution Partial Differential Equations

In the present thesis, we study global-in-time existence results, long-time decay estimates, and blow-up results for equations and systems of partial differential equations of evolution type. In particular, we consider dispersive and/or dissipative equations arising from physics, such as the wave equation with and without damping, the Boussinesq and Improved Boussinesq equations, Viscous Boussinesq, double dispersion models, the Klein-Gordon equation, models for the plate equation with mass or rotational inertia terms, and possible variants of these models involving fractional Laplace operators. For all these models, we investigate the role of oscillations in determining qualitative properties of solutions, such as time estimates of the solution to the corresponding linear problem. In this context, various tools of Harmonic Analysis are used, such as Fourier multipliers and the theory of oscillatory integrals. Sharp estimates for linear models are a fundamental tool for determining the critical exponent (or critical curve in the case of systems) of semilinear models associated with power-type nonlinearities. By critical exponent we mean the threshold exponent between the global existence of solutions for larger exponents, assuming sufficiently small initial data, and the non-existence of global solutions in the subcritical case, under suitable sign assumptions on the initial data. Throughout the thesis, we develop some new techniques for obtaining time estimates of the solution, using tools based on oscillatory integrals, such as the stationary phase method, and adapting them to various contexts: models in which the phase function, which is the function that determines the oscillations of the solution, is homogeneous of degree k, is a suitable perturbation of a homogeneous function of degree k, or presents degenerations at the level of the first or second derivative, under some symmetry hypothesis with respect to a suitable hypersurface. Furthermore, within the context of systems, we conveniently adapt the well-known test function method several times to obtain blow-up results in the case of systems with fractional powers of the Laplace operator, weak coupling with derivative-type nonlinearity, and other types of semilinear coupling. ​