The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable

Abstract

We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski's method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.

Well-posedness of hyperbolic systems with multiplicities and smooth coefficients

“…As observed in [14] combining the well-posedness results in [21,25] we already know that the Cauchy problem (1) is well-posed in the Gevrey class γ s , with 1 ≤ s < 1 + 1 m as well as in the corresponding spaces of (Gevrey-Beurling) ultradistributions. In this paper we want to prove that when A(t, D x ) has smooth coefficients and the condition (2) on the eigenvalues holds, then the Gevrey well-posedness result above can be extended to any s ≥ 1.…”

“…Finally, the matrix B(t, D x ) is made of three blocks of 3 × 9 matrices (21) via formula (23). More precisely, we get b…”

“…The general case. Recall from Section 3.4 that the m 2 × m 2 matrix B(t, ξ) is made up of m matrices of dimension m × m 2 that contain only in the last line non-zero elements, see (21). To not further complicate the notation, we will in what follows denote W (m) simply by W and will also assume that the b (l) ij (t, ξ) in B(t, ξ) are properly scaled by ξ l−m .…”

“…We mention here the work of D'Ancona, Kinoshita and Spagnolo [8] on weakly hyperbolic systems (i.e. systems with multiplicities) of size 2 × 2 and 3 × 3 with Hölder dependent coefficients later generalised to any matrix size by Yuzawa in [25] and to (t, x)-dependent coefficients by Kajitani and Yuzawa in [21]. In all these papers, well-posedness is obtained in Gevrey classes of a certain order depending on the regularity of the coefficients and the system size.…”

On hyperbolic equations and systems with non-regular time dependent coefficients

“…The dependence in x is a rather problematic issue and so far has been treated under strong regularity hypotheses (Gevrey). We mention the foundational work of Bronshtein [Bro80] and Nishitani [Nis83] for scalar equations and the paper of Kajitani and Yuzawa [KY06] for systems. The relationship between lower x-regularity and very weak solvability will be analysed in a future paper.…”