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

TL;DR

This paper establishes the well-posedness and analyzes the decay and regularity of solutions for a class of weakly hyperbolic Cauchy problems with variable multiplicities and unbounded coefficients, using advanced symbolic calculus.

Contribution

It introduces a parameter-dependent symbolic calculus and constructs a parametrix for weakly hyperbolic problems with variable multiplicities and unbounded coefficients.

Findings

Proves $ ext{S}( ext{R}^d)$-wellposedness of the Cauchy problem.

Provides results on global decay and regularity of solutions.

Develops a symbolic calculus tailored to variable multiplicity hyperbolic operators.

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