$L^p-L^q$ estimates for wave equations with strong time-dependent oscillations

Abstract

In this paper we consider the following Cauchy problem for the linear wave equation with time-dependent propagation speed: \begin{align} \label{Abstract.Cauchy} \tag{$\star$} \begin{cases} u_{tt}-\lambda^2(t)\omega^2(t)\Delta u=0, & (t,x)\in[0,\infty)\times \mathbb{R}^n, \\ u(0,x)=u_0(x), ~~ u_t(0,x)=u_1(x), & x\in\mathbb{R}^n, \end{cases} \end{align} where $\lambda=\lambda(t)$ is an increasing shape function and $\omega=\omega(t)$ is a bounded oscillating function which has very fast oscillations. The goal is to prove $L^p-L^q$ estimates on the conjugate line for Sobolev solutions of the Cauchy problem \eqref{Abstract.Cauchy} in the case that $\omega=\omega(t)$ has very fast oscillations. Basically, we apply the WKB analysis for the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. Then, the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter. This research continues the research of the paper [17].