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 R^d, 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(R^d)-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. (c) 2025 The Authors. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).