$L^2$-growth property for wave equations with higher derivative terms

TL;DR

This paper investigates the $L^2$-growth behavior of solutions to wave equations with higher derivative terms in low dimensions, providing sharp estimates and conditions under which growth occurs or is prevented.

Contribution

It derives sharp $L^2$-growth estimates for wave equations with higher derivatives in 1 and 2 dimensions and clarifies the conditions that inhibit growth in higher dimensions.

Findings

Sharp $L^2$-growth estimates in 1 and 2 dimensions.

Growth behavior does not occur in higher dimensions within the ($L^2 igcap L^1$) initial data framework.

Weighted $L^1$-initial velocity yields bounds on the solution.

Abstract

We consider the Cauchy problems in the whole space for wave equations with higher derivative terms. We derive sharp growth estimates of the $L^2$-norm of the solution itself in the case of the space 1, 2 dimensions. By imposing the weighted $L^1$-initial velocity, we can get the lower and upper bound estimates of the solution itself. In three or more dimensions, 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.