On hyperbolic systems with time-dependent Hölder characteristics

We want to study the Cauchy problem for first-order hyperbolic systems of the typewhere A and B are \(m\times m\) matrices of first-order and zero- order differential operators, respectively, with t-dependent coefficients, and u and \(g_0\) are column vectors with m entries. We work under the assumptions that the system matrix is of size \(m\times m\) with real eigenvalues and that the coefficients are of class \(C^{m-1}\) with respect to t. It follows that at the points of highest multiplicity the eigenvalues are of Hölder class \((m-1)/m\). We will therefore assume that the matrix \(A(t,\xi )\) has m real eigenvalues \(\lambda _j(t,\xi )\) of Hölder class \(C^\alpha \), \(0<\alpha \le 1\) with respect to t. Note that it is not restrictive to assume that the eigenvalues \(\lambda _j\), \(j=1,\ldots ,m\), are ordered because we can always reorder them to satisfy this (ordering) assumption, and the Hölder continuity is preserved by such reordering. If \(\alpha =1\), it is sufficient to assume that \(\lambda _j\), \(j=1,\ldots ,m\), are Lipschitz.

In analogy with scalar equations in [3, 7], we work under the hypothesis of the following uniform property: There exists a constant \(c>0\) such thatfor all \(1\le i,j,k\le m\), \(t\in [0,T]\) and \(\xi \in \mathbb {R}^n\).

The research of this paper continues investigations of properties of solutions to Cauchy problems for hyperbolic equations with multiplicities. The case of time-dependent coefficients already presents a number of challenging problems, most importantly in view of the fact that already the scalar wave equationin dimension \(n=1\) may not be well-posed even for smooth data \(v_0,v_1\in C^\infty \). More precisely, if \(a\in C^\alpha \) is Hölder with \(0<\alpha <1\), even in the strictly hyperbolic case \(a>0\) the Cauchy problem (3) may have nonunique solutions (see Colombini et al. [2]), and in the weakly hyperbolic case \(a\ge 0\) even if a is smooth \(a\in C^\infty \), the Cauchy problem (3) may have no distributional solutions (see Colombini and Spagnolo [4]). However, the Cauchy problem (3) is well-posed in suitable Gevrey classes (see Colombini et al. [1]). At the moment, scalar higher-order equations with time-dependent coefficients are relatively understood (see, e.g. [3, 13] and their respective extensions in [7, 8]). Further extreme cases: analytic coefficients and distributional coefficients have been also investigated (see, e.g. authors’ papers [6, 9, 10], respectively, and references therein). Hyperbolic systems of the form (1) have been also investigated (see, e.g. Garetto [6], Kajitani and Yuzawa [14] and Yuzawa [15]).

The main new idea behind this paper enabling us to obtain an improvement in the well-posedness results for the system in (1) is the transformation of the system (1) to a larger system which, however, 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 \(C^{m-1}\) with respect to t, is carried out following the method of D’Ancona and Spagnolo [5], leading to the Cauchy problem of the formThis is a Cauchy problem for the first-order hyperbolic system of the size \(m^2\times m^2\) of pseudo-differential equations. Despite the increase in the size of the system from \(m\times m\) to \(m^2\times m^2\) and the change from a differential system to a pseudo-differential one, the system (4) has a crucial advantage of being in a block Sylvester form (see (9) for a precise formulation). This allows us to implement the ideas developed in [7] for scalar equations, where the reduction in a scalar equation to a Sylvester form system was performed.

To summarise our result, here we first note that combining the results in [14, 15] we already know that the Cauchy problem (1) is well-posed in the Gevrey class \(\gamma ^s\) , withArguing by the Fourier characterisation of Gevrey–Beurling ultradistributions, one can easily extend the Gevrey well-posedness above to spaces of ultradistributions. It is our aim in this paper to show that the interval of Gevrey well-posedness in (5) can be enlarged under the uniform property (2) of the eigenvalues. Since by the results of Kajitani and Yuzawa at least an ultradistributional solution exists for Gevrey initial data with \(s\ge 1+\frac{\alpha }{m}\) we will prove, for suitable values of s, that this solution is indeed Gevrey, because it solves the reduced Cauchy problem (4). In this sense, the well-posedness of (1) can be determined by studying the well-posedness of the reduced Cauchy problem (4). More precisely, by standard arguments it is sufficient to find an a priori estimate on the Fourier transform with respect to x of the solution U of (4).

We assume that the Gevrey classes \(\gamma ^s({\mathbb R}^n)\) are well known: These are spaces of all \(f\in C^\infty (\mathbb {R}^n)\) such that for every compact set \(K\subset \mathbb {R}^n\) there exists a constant \(C>0\) such that for all \(\beta \in \mathbb N_0^n\) we have the estimateFor \(s=1\), we obtain the class of analytic functions. We refer to [7] for a detailed discussion and Fourier characterisations of Gevrey spaces of different types. Since we are dealing with vectors in this paper, we will write \(\gamma ^s({\mathbb R}^n)^m\) for m-vectors consisting of functions in \(\gamma ^s({\mathbb R}^n)\). This is our main result:

The energy estimates obtained in the proof of Theorem 1.1 allow one to also obtain the ultradistributional well-posedness results. First we note that the Gevrey spaces \(\gamma ^s(\mathbb {R}^n)\) considered in (6) are of Gevrey–Roumeau type. At the same time, we denote by \(\gamma ^{(s)}(\mathbb {R}^n)\) the Gevrey spaces of Gevrey–Beurling type, i.e. the space of all \(f\in C^\infty ({\mathbb R}^n)\) such that for every compact set \(K\subset \mathbb {R}^n\) and for every constant \(A>0\) there exists a constant \(C_{A,K}>0\) such that for all \(\beta \in \mathbb N_0^n\) we have the estimateFor \(1<s<\infty \), we denote by \(\mathcal D_{(s)}^\prime (\mathbb {R}^n):=(\gamma ^{(s)}_c(\mathbb {R}^n))'\) the topological dual of compactly supported functions in \(\gamma ^{(s)}(\mathbb {R}^n)\) and by \({\mathcal E}'_{(s)}(\mathbb {R}^n)\) the topological dual of \(\gamma ^{(s)}(\mathbb {R}^n)\). Consequently, arguing similarly to [7], the proof of Theorem 1.1 yields the following ultradistributional well-posedness: