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

Abstract

We consider the Cauchy problem in $R^n$ for the wave equation with a higher derivative term. We derive sharp growth estimates of the $L^2$-norm of the solution itself for the case of $n=1$ and $n=2$. By imposing the weighted $L^1$-initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of $n\ge 3$, we observe that the $L^2$-growth behavior of the solution never occurs in the $(L^2\cap L^1)$-framework of the initial data.