Asymptotic profiles of solutions for structural damped wave equations

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 $0<\sigma \le 1$. 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 $\sigma$.