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 $\mathscr{ S}(\mathbb{ R}^{ d})$-wellposedness and give results about the global decay and regularity of the solution, within a scale of weighted Sobolev space.
Categories: math.AP