THE CAUCHY PROBLEM FOR WAVE EQUATIONS WITH NON LIPSCHITZ COEFFICIENTS; APPLICATION TO CONTINUATION OF SOLUTIONS OF SOME NONLINEAR WAVE EQUATIONS

In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form Lu := Sigma(n)(j,k=0) partial derivative(yj) (a(j,k)partial derivative(yk)u) + Sigma(n)(j=0){b(j)partial derivative(yj) u+partial derivative(yj)(c(j)u)} + du = f, when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N. Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on u: the smooth solution can be extended as long as it remains Log-Lipschitz.