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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 thetime variable is governed by a suitable "shape function". We develop aparameter-dependent symbolic calculus, corresponding to an appropriatesubdivision of the phase space. By means of such calculus, a parametrix can beconstructed, in terms of (generalized) Fourier integral operators naturallyassociated with the employed symbol class. Further, employing the parametrix,we prove $\mathscr{S}(\mathbb{R}^{d})$-wellposedness and give results about theglobal decay and regularity of the solution, within a scale of weighted Sobolevspace.

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