Asymptotic profile of solutions to the double dispersion equation

In this paper, we investigate the initial value problem for the double dispersion equation in RnRn. Decay estimate and asymptotic profile of solutions to the corresponding linear problem are established. The asymptotic profile of solutions to the corresponding linear problem is given by the convolution of the fundamental solutions of heat and free wave equations. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear problem obtained in this paper is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to the nonlinear problem are also established.