On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: "smoothness", "oscillations", "degeneration" and "stabilization". Gevrey and C-infinity well-posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim