Estimates for a class of oscillatory integrals and decay rates for wave-type equations

Abstract

This paper investigates higher order wave-type equations of the form ${\partial }_{tt}u+P(D_x)u=0$, where the symbol $P(\xi )$ is a real, non-degenerate elliptic polynomial of the order $m\ge 4$ on ${\mathbf{R}}^n$. Using methods from harmonic analysis, we first establish global pointwise time-space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) $L^p-L^q$ estimates on the wave solution in terms of the initial conditions.

References in corpus (4)

• Global Strichartz estimates for nontrapping perturbations of the Laplacian
• Global estimates of fundamental solutions for higher-order Schrödinger equations
• Scattering for the Beam equation in low dimensions
• Convex Hypersurfaces and $L^p$ Estimates for Schrödinger Equations