$L^2$ asymptotic profiles of solutions to linear damped wave equations

TL;DR

This paper derives higher order asymptotic profiles for solutions to the linear damped wave equation in multi-dimensional space, revealing that the hyperbolic part's expansion order varies with spatial dimension.

Contribution

It introduces a novel hyperbolic asymptotic expansion for the damped wave equation, with the expansion order depending on the spatial dimension, which is a new insight.

Findings

Higher order asymptotic profiles are obtained.

The hyperbolic part's expansion order depends on the spatial dimension.

The results extend understanding of solution behavior in damped wave equations.

Abstract

In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $\textbf{R}^n$ \begin{equation*} u_{tt}-\Delta u+u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x), \end{equation*} where $n\in\textbf{N}$ and $u_0$, $u_1\in L^2(\textbf{R}^n)$. Established hyperbolic part of asymptotic expansion seems to be new in the sense that the order of the expansion of the hyperbolic part depends on the spatial dimension.