Asymptotic profile of solutions for semilinear wave equations with structural damping

TL;DR

This paper investigates the global existence and asymptotic behavior of solutions to a semilinear wave equation with structural damping, showing solutions tend to a fundamental parabolic solution under certain conditions.

Contribution

It establishes global existence for small initial data and characterizes the asymptotic profile as a multiple of the fundamental solution of the related parabolic equation.

Findings

Global existence for small initial data in weighted Sobolev spaces.

Asymptotic profile converges to a fundamental parabolic solution.

Results hold for dimensions n ≥ 2 and damping parameter σ in (0, 1/2).

Abstract

This paper is concerned with the initial value problem for semilinear wave equation with structural damping $u_{tt}+(-\Delta)^{\sigma}u_t -\Delta u =f(u)$, where $\sigma \in (0,\frac{1}{2})$ and $f(u) \sim |u|^p$ or $u |u|^{p-1}$ with $p> 1 + {2}/(n - 2 \sigma)$. We first show the global existence for initial data small in some weighted Sobolev spaces on $\mathcal R^n$ ($n \ge 2$). Next, we show that the asymptotic profile of the solution above is given by a constant multiple of the fundamental solution of the corresponding parabolic equation, provided the initial data belong to weighted $L^1$ spaces.