Asymptotic profiles of solutions for structural damped wave equations

TL;DR

This paper studies the long-term behavior of solutions to structurally damped wave equations, classifying their asymptotic profiles into five patterns based on damping parameters.

Contribution

It provides a detailed classification of asymptotic solution profiles for a class of damped wave equations, extending understanding of their long-time dynamics.

Findings

Solutions asymptotically resemble specific functions as time goes to infinity.

Five distinct asymptotic profile patterns are identified based on parameters.

Approximation formulas for solutions are derived for large time.

Abstract

In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $\partial_{t}^{2} u - \Delta u + \nu (-\Delta)^{\sigma} \partial_{t} u=0$, where $\nu >0$ and $0< \sigma \le1$. Our result is the approximation formula of the solution by a constant multiple of a special function as $t \to \infty$, which states that the asymptotic profiles of the solutions are classified into $5$ patterns depending on the values $\nu$ and $\sigma$.