Abstract
In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in [1]. In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.
Highlights
We continue the study of non-diagonalisable systems that begun in [1] by proving a result on solution representations and propagation of singularities for a hyperbolic system with x-dependent principal part
In this paper we will use the short expression integrated Fourier integral operator to denote an operator of the type t eiφ(t,s,x,ξ)a(t, s, x, ξ )f (ξ, s)dξ ds, 0 Rn where the Fourier transform of f = f (y, s) is meant with respect to the variable y
This section contains some auxiliary results on Fourier integral operators and related integral operators that we will use throughout the paper
Summary
We continue the study of non-diagonalisable systems that begun in [1] by proving a result on solution representations and propagation of singularities for a hyperbolic system with x-dependent principal part. In [8], Rozenblum considered smoothly diagonalisable systems with transversally intersecting characteristics, and derived a formula for the propagation of its singularities. The transversality condition was removed in [7], replaced by a weaker condition of intersection of finite order at points of multiplicity, with propagation of singularities result as well. We extend the results of [7] to non-diagonalisable hyperbolic systems with variable multiplicity.
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