Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

Abstract

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.

Well‐posedness and scattering of solutions to the sixth‐order Boussinesq type equation in the framework of modulation spaces

“…Very recently, Li et al investigated the initial boundary value problem of on a half line with nonhomogeneous boundary conditions and proved the problem is locally well‐posed in for . For the study of other Boussinesq type equation, we refer to Wang and Su, Wang, Wang and Wang, and Xu et alDe GodefroyEsfahani and LevandoskyWang and Esfahani…”

Asymptotic Profiles and Convergence Rates of the Linearized Compressible Navier–Stokes– Korteweg System

“…As far as we know, there are few results about asymptotic profiles of solutions to the linearized compressible Navier-Stokes-Korteweg system (2). In this paper, our main aim is to establish the asymptotic profiles of solutions to the problems (2) and (3) in the spirit of [17][18][19][20][21]. More precisely, we show that the asymptotic profile of solutions is given by the convolution of the fundamental solutions of diffusion and free wave equations.…”