Global Wellposedness of a Class of Weakly Hyperbolic Cauchy Problems with Variable Multiplicities on ${\mathbb{R}}^d$

Abstract

We study a class of weakly hyperbolic Cauchy problems on ${\mathbb{R}}^d$, involving linear operators with characteristics of variable multiplicities, whose coefficients are unbounded in the space variable. The behaviour 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 $\mathcal{S}({\mathbb{R}}^d)$-wellposedness and give results about the global decay and regularity of the solution, within a scale of weighted Sobolev space.

52 pages. Misprints and other minor corrections. Extension and improvement of some arguments and examples