# On hyperbolic systems with time-dependent Hölder characteristics

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## 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).

## Keywords

Hyperbolic equations Gevrey spaces Ultradistributions## Mathematics Subject Classification

Primary 35L25 35L40 Secondary 46F05## 1 Introduction

*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.

*uniform property*: There exists a constant \(c>0\) such that

Assumptions of Hölder regularity of this type and the uniform condition (2) are rather natural (see Colombini and Kinoshita [3] and the authors’ paper [7] for a discussion and examples). In particular, Colombini and Kinoshita [3] treated the scalar version of the Cauchy problem (1) with \(n=1\), and the authors extended it to the multidimensional case \(n\ge 1\) in [7], also improving some Gevrey indices.

*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]).

*t*, is carried out following the method of D’Ancona and Spagnolo [5], leading to the Cauchy problem of the form

*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).

*m*-vectors consisting of functions in \(\gamma ^s({\mathbb R}^n)\). This is our main result:

### **Theorem 1.1**

*A*and

*B*are of class \(C^{m-1}\) and 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*, that satisfy (2). Let \(T>0\) and \(g_0\in \gamma ^s({\mathbb R}^n)^m\). Then, the Cauchy problem (1) has a unique solution \(u\in C^1([0,T],\gamma ^s({\mathbb R}^n)^m)\) provided that

For the proof, we can assume that \(s>1\) since the case \(s=1\) is essentially known (see [11, 12]).

Also, we note that the proof also covers the case \(\alpha =1\), in which case it is enough to assume that the eigenvalues are Lipschitz.

We note that the result of Theorem 1.1 is an improvement of known results in terms of the Gevrey order. For example, this is an improvement of Yuzawa’s and Kajitani’s order (5) from [14, 15]. See Remark 2.3 for more details.

### **Theorem 1.2**

*A*and

*B*are of class \(C^{m-1}\) and 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*, that satisfy (2). Let \(T>0\) and \(g_0\in ({\mathcal E}_{(s)}^\prime (\mathbb {R}^n))^m\). Then, the Cauchy problem (1) has a unique solution \(u\in C^1([0,T],(\mathcal D_{(s)}^\prime (\mathbb {R}^n))^m)\) provided that

## 2 Proof of Theorem 1.1

The first step in our new approach to the Cauchy problem (1) is to rewrite the system in a special form, i.e. in block Sylvester form. This is possible thanks to the reduction given by D’Ancona and Spagnolo [5], which is summarised in the following subsection.

### 2.1 Reduction to block Sylvester form

*I*is the \(m\times m\) identity matrix. By applying the corresponding operator \(L(t,D_t,D_x)\) to (1), we transform the system

*A*and

*B*are of class \(C^{m-1}\) with respect to

*t*the equation above has continuous

*t*-dependent coefficients. Indeed, the coefficients of the equation \(D_t u-A(t,D_x)u-B(t)u=0\) are of class \(C^{m-1}\) and the operator \(L(t,D_t,D_x)\) is of order \(m-1\) being defined via the cofactor matrix of a \(m\times m\) matrix. Note that \(\delta (t,D_t,D_x)\) is the operator

*m*which can be transformed into a first-order system of size \(m^2\times m^2\) of pseudo-differential equations, by setting

*m*identical blocks of the type

*m*blocks of size \(m\times m^2\) of the type

*u*is the solution of the Cauchy problem (1) with \(u(0,x)=g_0\). As observed in the introduction, we already know that such

*u*exists at least as ultradistribution. By using the initial condition \(g_0\) and by deriving the system in (1) \(m-1\) times with respect to

*t*, we obtain that \(D_t^{j-1}\langle D_x \rangle ^{m-j}u(0,x)\) has the same regularity properties of \(g_0\) for \(j=1,\ldots ,m\). It follows that if \(g_0\in \gamma ^s(\mathbb {R}^n)^m\) then \(U_0\in \gamma ^s(\mathbb {R}^n)^{2m}\).

### 2.2 Energy estimates

*t*and we separate them by adding some power of a parameter \(\varepsilon \rightarrow 0\). In detail, assuming that the \(\lambda _j\)’s are ordered and taking a mollifier \(\varphi \in \mathcal {C}^\infty _{\text {c}}(\mathbb {R})\), \(\varphi \ge 0\) with \(\int \varphi (t)\, dt=1\) we set

### **Proposition 2.1**

Let \(\varphi \in C^\infty _{c}(\mathbb {R})\), \(\varphi \ge 0\) with \(\int _\mathbb {R}\varphi (x)\, dx=1\).

- (i)
\(|\partial _t\lambda _{j,\varepsilon }(t,\xi )|\le c\,\varepsilon ^{\alpha -1}\langle \xi \rangle \),

- (ii)
\(|\lambda _{j,\varepsilon }(t,\xi )-\lambda _j(t,\xi )|\le c\,\varepsilon ^{\alpha }\langle \xi \rangle \),

- (iii)
\(\lambda _{j,\varepsilon }(t,\xi )-\lambda _{i}(t,\xi )\ge \varepsilon ^\alpha \langle \xi \rangle \) for \(j>i\),

*m*identical blocks of the type

*x*as well. Hence, instead of dealing with the Cauchy problem (9) directly we can apply the Fourier transform with respect to

*x*to it and focus on the corresponding Cauchy problem

- (i)
\(\frac{\partial _t\det H_\varepsilon }{\det H_\varepsilon }\),

- (ii)
\(\Vert H_\varepsilon ^{-1}\partial _t H_\varepsilon \Vert \),

- (iii)
\(\Vert H_\varepsilon ^{-1}\mathcal {A}H_\varepsilon -(H_\varepsilon ^{-1}\mathcal {A}H_\varepsilon )^*\Vert \),

- (iv)
\(\Vert H_\varepsilon ^{-1}\mathcal {L}H_\varepsilon -(H_\varepsilon ^{-1}\mathcal {L}H_\varepsilon )^*\Vert \).

#### 2.2.1 Estimate of (i), (ii), (iii) and (iv)

*m*identical blocks of the \(m^2\times m^2\)-matrix \(H_\varepsilon \) are exactly given by the matrix

*H*used in the paper [7] (formula (3.4)). Hence, we can set

*m*identical blocks \(H^{-1}\) as defined in Proposition 17(ii) in [7]. It follows that to estimate \(\Vert H_\varepsilon ^{-1}\partial _t H_\varepsilon \Vert \) it is enough to estimate the norm of the corresponding block \(H^{-1}\partial _t H\). This has been done in Subsection 4.2 in [7] and leads to

*m*blocks of the type \(H^{-1}\mathcal {A}H-(H^{-1}\mathcal {A}H)^*\). This is the type of matrix which has been estimated in Subsection 4.3 in [7]. In detail, \(\Vert H^{-1}\mathcal {A}H-(H^{-1}\mathcal {A}H)^*\Vert \le c_3\varepsilon ^\alpha \langle \xi \rangle \) and therefore

*m*blocks of the type \((\mathrm{det}H)^{-1}\) times a matrix with 0-order symbols bounded with respect to \(\varepsilon \) (see Subsection 4.4. in [7]). More precisely, by following the arguments of Proposition 17(iv) in [7], we get the estimate

### 2.3 Conclusion of the proof of Theorem 1.1

### *Remark 2.2*

We have proven that the solution *u* of the Cauchy problem (1) is of class \(C^1\) with respect to *t*. Since the coefficients of the matrices *A* and *B* are of class \(C^{m-1}\), it actually follows that *u* belongs to \(C^m([0,T];\gamma ^s(\mathbb {R}^n)^m)\).

### *Remark 2.3*

*t*,

*x*)-dependent systems in [14] have proven well-posedness in the Gevrey class \(\gamma ^s\), with

### *Remark 2.4*

The strategy adopted in the proof of Theorem 1.1 shows how the energy estimate used for scalar equations in [7] can be directly applied to systems after reduction to block Sylvester form to obtain Gevrey well-posedness. In the same way, one can get well-posedness in spaces of ultradistributions. In other words, Theorem 1.2 is proven by arguing on the reduced Cauchy problem (9) as in Subsection 4.5 from the aforementioned paper.

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