L2$L^{2}$‐growth property for the wave equation with a higher derivative term

Abstract

AbstractWe consider the Cauchy problem in for the wave equation with a higher derivative term. We derive sharp growth estimates of the ‐norm of the solution itself for the case of and . By imposing the weighted ‐initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of , we observe that the ‐growth behavior of the solution never occurs in the ‐framework of the initial data.