On hyperbolic systems with time-dependent H\xf6lder characteristics

Claudia Garetto,Michael Ruzhansky

doi:10.1007/s10231-016-0567-6

Abstract

In this paper, we study the well-posedness of weakly hyperbolic systems with time-dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to t. In the past, these kinds of systems have been investigated by Yuzawa (J Differ Equ 219(2):363–374, 2005) and Kajitani and Yuzawa (Ann Sc Norm Super Pisa Cl Sci (5) 5(4):465–482, 2006) by employing semigroup techniques (Tanabe–Sobolevski method). Here, under a certain uniform property of the eigenvalues, we improve the Gevrey well-posedness result of Yuzawa (2005) and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the system to block Sylvester form and then the formulation of suitable energy estimates inspired by the treatment of scalar equations in Garetto and Ruzhansky (J Differ Equ 253(5):1317–1340, 2012).

Highlights

We want to study the Cauchy problem for first-order hyperbolic systems of the typeDt u − A(t, Dx )u − B(t)u = 0, x ∈ Rn, t ∈ [0, T ], (1)u|t=0 = g0, where A and B are m × m matrices of first-order and zero- order differential operators, respectively, with t-dependent coefficients, and u and g0 are column vectors with m entries
We work under the assumptions that the system matrix is of size m × m with real eigenvalues and that the coefficients are of class Cm−1 with respect to t
It follows that at the points of highest multiplicity the eigenvalues are of Hölder class (m − 1)/m

Summary

Introduction

We want to study the Cauchy problem for first-order hyperbolic systems of the type. u|t=0 = g0, where A and B are m × m matrices of first-order and zero- order differential operators, respectively, with t-dependent coefficients, and u and g0 are column vectors with m entries. The main new idea behind this paper enabling us to obtain an improvement in the wellposedness results for the system in (1) is the transformation of the system (1) to a larger system which, enjoys the property of being in block Sylvester form Such a transformation, which can be performed under the assumption that the system coefficients are of class Cm−1 with respect to t, is carried out following the method of D’Ancona and Spagnolo [5], leading to the Cauchy problem of the form. Since we are dealing with vectors in this paper, we will write γ s (Rn)m for m-vectors consisting of functions in γ s(Rn) This is our main result: Theorem 1.1 Assume that coefficients of the m × m matrices A and B are of class Cm−1 and that the matrix A(t, ξ ) has m real eigenvalues λ j (t, ξ ) of Hölder class Cα, 0 < α ≤ 1 with respect to t, that satisfy (2).

Hε W

It follows that