On Linear Homogeneous Biwave Equations

Abstract

The biwave maps are a class of fourth order hyperbolic equations. In this paper, we are interested in the solution formulas of the linear homogeneous biwave equations. Based on the solution formulas and the weighted energy estimate, we can obtain the $L^\infty(\mathbb R^n)-W^{N,1}(\mathbb R^n)$ and $L^\infty(\mathbb R^n)-W^{N,2}(\mathbb R^n)$ estimates, respectively. By our results, we find that the biwave maps enjoy some different properties compared with the standard wave equations.