Local in Space Energy Estimates for Second-order Hyperbolic Equations

Abstract

In the first part of the paper we review some recent results concerning the propagation of analytic regularity for the s-Gevrey solutions, with $s < \overline{m}/(\overline{m}-1)$, to the semilinear (weakly) hyperbolic equations with characteristics of multiplicity $\leq\overline{m}$.The main results are concerning two special classes of equations: the equations with coefficients depending only on time, and those in space dimension one. These results rely on suitable a priori estimates for the corresponding linearized equations, based on the theory of quasisymmetrizer. In view of these a priori estimates, the case of several space variables is quite different from that of one space variable: in the latter the quasisymmetrizer provides an energy integral on each open subset of $\mathbb{R}^1$, and this ensures the propagation of regularity along the cones of determinacy, whereas in the multidimensional case we can only define the energy on the whole $\mathbb{R}^n$

The second part of the paper is devoted to the Cauchy problem for the second-order linear hyperbolic equations. For these equations, we are in the position to get an energy estimate along the cones of determinacy, which will imply, for the corresponding semilinear equations, the analytic propagation along these cones for all the s-Gevrey solution with $s < 2$. The proof of this energy estimate is only sketched.

Mathematics Subject Classification. Primary 35L15; Secondary 35B65.