Loss of regularity for the solutions to hyperbolic equations with non‐regular coefficients—an application to Kirchhoff equation

We consider the Cauchy problem for second‐order strictly hyperbolic equations with time‐depending non‐regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C^∞ well‐posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd.