Global well-posedness of a class of weakly hyperbolic Cauchy problems with variable multiplicities on R^d

We study a class of weakly hyperbolic Cauchy problems on Rd, involving linear operators with characteristics of variable multiplicities, whose coefficients are unbounded in the space variable. The behavior in the time variable is governed by a suitable “shape function”. We develop a parameter-dependent symbolic calculus, corresponding to an appropriate subdivision of the phase space. By means of such calculus, a parametrix can be constructed, in terms of (generalized) Fourier integral operators naturally associated with the employed symbol class. Further, employing the parametrix, we prove S(Rd)-well-posedness and give results about the global regularity of the solution, within a scale of weighted Sobolev space, encoding both smoothness and decay at infinity of temperate distributions. In particular, loss of decay appears, together with the well-known phenomenon of loss of smoothness