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.