On the Cauchy problem for a class of hyperbolic operators with triple characteristics

Abstract

The Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, an existence result for the Cauchy problem is obtained.

1 Introduction

In the past, many authors studied widely hyperbolic operators with double characteristics, both in the case when there is no transition between different types on the set where the principal symbol vanishes of order 2 (see for instance [5, 8] for a general survey) and when there is transition (see [1,2,3,4]). The operators are called effectively hyperbolic if the propagation cone C is transversal to the manifold of multiple points (see [8]). Moreover, if this occurs and lower order terms satisfy a generic Ivrii-Petkov vanishing condition, we have well posedness in $C^{\infty }$ (see [7]).

The aim of the paper is to analyze the following class of operators with triple characteristics

$$\begin{aligned} P(x_0, D)= D^3_{x_0} - (D^2_{x_1} + x_1^2 D_{x_2}^2) D_{x_0} - b x_1^3 D^3_{x_2}, \quad \mathrm{in} \ \varOmega = ]0, + \infty [ \times {\mathbb {R}}^2, \end{aligned}$$

where $D_{x_j}=\frac{1}{i}\partial _{x_{j}},j=0,1,2,$ under hyperbolicity assumptions, namely $|b| \le \frac{2}{3}$. Such a class of operators has been considered in [6], for example operators whose propagation cone is not transversal to the triple characteristic manifold. The authors prove a well posedness result in the Gevrey category for a simple hyperbolic operator with triple characteristics and whose propagation cone is not transversal to the triple manifold. Furthermore they estimate the precise Gevrey threshold, by exhibiting a special class of solutions, through which we can violate weak necessary solvability conditions. More precisely, let $x=(x_0,x')$ where $x'=(x_1,x_2)$, let $\xi =(\xi _0, \xi ')$, where $\xi '= (\xi _1, \xi _2)$. In [6], the authors study the well posedness of the following Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Pu= 0, \quad \mathrm{in} \ \varOmega =]0, + \infty [ \times {\mathbb {R}}^2, \\ D_{x_j}u(0,x')=\phi _j(x'), \ j=0,1,2, \end{array}\right. } \end{aligned}$$

with $\phi _j(x') \in \gamma ^{(s)} ({\mathbb {R}}^2)$, $j=0,1,2$, where $\gamma ^{(s)} ({\mathbb {R}}^2)$ is the Gevrey s class. They obtained that the Cauchy problem for P is well posed in the Gevrey 2 class assuming that $b^2 < \frac{4}{27}$. Moreover, if $s > 2$, it is possible to choose $b \in \left]0, \frac{2}{3 \sqrt{3}} \right[$ such that the Cauchy problem for P is not locally solvable at the origin in the Gevrey s class.

In this paper, instead, we investigate on the well posedness of the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Pu= f, \quad \mathrm{in} \ \varOmega , \\ D_{x_j}u(0,x')=0, \ j=0,1,2, \end{array}\right. } \end{aligned}$$

(1)

with $f \in H^r(\varOmega )$, in the Sobolev spaces, obtaining an existence result for solutions.

Let us set

$$\begin{aligned} Q= - \partial ^3_{x_0} + \left( \partial ^2_{x_1} + x_1^2 \partial _{x_2}^2\right) \partial _{x_0} + b x_1^3 \partial ^3_{x_2}, \quad \mathrm{in} \ \varOmega . \end{aligned}$$

It results

$$\begin{aligned} Pu= iQu, \quad \mathrm{in} \ \varOmega . \end{aligned}$$

As a consequence, problem (1) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} Qu= g, \quad \mathrm{in} \ \varOmega , \\ \partial _{x_j}u(0,x')=0, \ j=0,1,2, \end{array}\right. } \end{aligned}$$

(2)

where we set $g = i f$, in $\varOmega$, with g real function. The main result of the paper is the following.

Theorem 1

Let $f \in H^r_{loc}({\overline{\varOmega }})$, with $r \ge 5$. For every $h, T>0$, the Cauchy problem (1) admits a solution $u \in H^{r-2}(\varOmega _{h,T})$, where $\varOmega _{h,T} = [0,h[ \times ]-T,T[^2$.

The rest of the paper is organized as follows. Section 2 deals with some preliminary notations and definitions. In Sect. 3 some a priori estimates are established. Section 4 is devoted to obtain a priori estimates in Sobolev spaces with negative indexes. Finally, the existence result for solutions to the Cauchy problem are proved in Sect. 5.

2 Notations and preliminaries

Let $\alpha =(\alpha _0, \alpha _1, \alpha _2) \in {\mathbb {N}}^3_0$. Let $\partial ^{\alpha }$ be the derivative of order $|\alpha |$, let $\partial ^{h}_{x_j}$ be the derivative of order h with respect to $x_j$ and let $\partial ^{h}_{x_j, x_p}$ be the derivative of order h with respect to $x_j$ and $x_p$.

We indicate the $L^2$-scalar product, the $L^2$-norm and the $H^{r}$-norm ($r\in \mathbb {N}_0$) by $(\cdot , \cdot )$, $\Vert \cdot \Vert$ and $\Vert \cdot \Vert _{H^{r}}$ respectively.

Let $\varOmega$ be an open subset of ${\mathbb {R}}^3$. Let $C_0^{\infty }({\overline{\varOmega }})$ be the space of the restrictions to ${\overline{\varOmega }}$ of functions belonging to $C^{\infty }_0({\mathbb {R}}^3)$. For each $K \subseteq {\overline{\varOmega }}$ compact set, let $C^{\infty }_0 (K)$ be the set of functions $\varphi \in C^{\infty }_0 ({\overline{\varOmega }})$ having support contained in K. Let $S({\mathbb {R}}^3)$ be the space of rapidly decreasing functions. In particular, let $S({\overline{\varOmega }})$ be the space of the restrictions to ${\overline{\varOmega }}$ of functions belonging to $S({\mathbb {R}}^3)$.

Let $\varOmega = [0, + \infty [ \times ]a_1,b_1[ \times ]- \infty , + \infty [$ and let $s \in {\mathbb {R}}$, let us denote by $\Vert \cdot \Vert _{H^{0,0,s}({\overline{\varOmega }})}$ the norm given by

$$\begin{aligned} \Vert u \Vert ^2_{H^{0,0,s}({\overline{\varOmega }})}= & {} \int _0^{+ \infty } dx_0 \int _{a_1}^{b_1} dx_1 \\&\quad \int _{- \infty }^{+ \infty }\frac{1}{2 \pi } (1+|\xi _2|^2)^{s} | {\widehat{u}}(x_0, x_1, \xi _2)|^2 d\xi _2, \quad \forall u \in C^{\infty }_{0}({\overline{\varOmega }}), \end{aligned}$$

where the Fourier transform is performed only with respect to the variable $x_2$. Moreover, let us denote by $A_s$ the pseudodifferential operator given by

$$\begin{aligned} A_s u(x)= \int _{- \infty }^{+ \infty } \frac{1}{2 \pi } e^{i x_2 \cdot \xi _2} (1+|\xi _2|^2)^{\frac{s}{2}} {\widehat{u}}(x_0, x_1, \xi _2) d\xi _2, \quad \forall u \in C^{\infty }_0({\overline{\varOmega }}). \end{aligned}$$

(3)

Let us recall that $A_s: C^{\infty }_0({\overline{\varOmega }}) \rightarrow C^{\infty }({\overline{\varOmega }})$. For every $\varphi (x_2) \in C^{\infty }_0({\mathbb {R}})$, the operator $\varphi A_s u$ extends to a linear continuous operator from $H^{0,0,r}_{comp.}({\overline{\varOmega }})$ to $H^{0,0,r-s}_{loc}({\overline{\varOmega }})$, where $r,s \in {\mathbb {R}}$. In particular, in $\varOmega _k=[0,k[ \times ]a_1,b_1[ \times ]- \infty , + \infty [$, for $k>0$, we denote by $H^{0,0,s}(\varOmega _k)$ the space of all $u \in H^{0,0,s}(\varOmega )$ such that $\mathrm{supp} \; u \subseteq \varOmega _k$. Moreover, denoted by ${\mathcal {U}}_{x_2}$ the projection of $\mathrm{supp} \ u$ on the axis $x_2$, if $\mathrm{supp} \ \varphi \subseteq {\mathbb {R}} \backslash {\mathcal {U}}_{x_2}$, then $\varphi A_s u$ is regularizing with respect to the variable $x_2$, namely it results:

$$\begin{aligned} \Vert \varphi A_s u \Vert _{H^{0,0,r}} \le c \Vert u \Vert _{H^{0,0,r'}}, \quad \forall r,r' \in {\mathbb {R}}, \ u \in C^{\infty }({\overline{\varOmega }}). \end{aligned}$$

The norms $\Vert u \Vert _{H^{0,0,s}(\varOmega )}$ and $\Vert A_s u \Vert _{L^2(\varOmega )}$ are equivalent for any $s \in {\mathbb {R}}$.

Let $s \in {\mathbb {R}}$ and $p \ge 0$. Let $H^{p,s}({\mathbb {R}}^3)$ be the space of distributions U into ${\mathbb {R}}^3$ such that

$$\begin{aligned} \Vert u \Vert ^2_{H^{p,s}({\mathbb {R}}^3)} = \frac{1}{2 \pi } \sum _{|h| \le p} \int _{{\mathbb {R}}^3} (1+|\xi _2|^2)^{s} | \partial ^h_{x_0,x_1} {\widehat{U}}(x_0, x_1, \xi _2)|^2 dx_0 dx_1 d\xi _2 < + \infty . \end{aligned}$$

At last, let $H^{p,s}(\varOmega )$ be the space of the restrictions to $\varOmega$ of elements of $H^{p,s}({\mathbb {R}}^3)$ endowed with the norm

$$\begin{aligned} \Vert u \Vert _{H^{p,s}(\varOmega )} = \inf _{{\begin{array}{c} U \in H^{p,s}({\mathbb {R}}^3) \\ U|_{\varOmega } =u \end{array}}} \Vert U \Vert _{H^{p,s}({\mathbb {R}}^3)}. \end{aligned}$$

3 A priori estimates

The following preliminary result holds (see [1], Lemma 3.1).

Lemma 1

Let $u \in S({\overline{\varOmega }})$ and let $p, \alpha _0, \alpha _1, \alpha _2 \in {\mathbb {N}}_0$. Then

$$\begin{aligned} \Vert x_0^{\frac{p}{2}} \partial ^{\alpha _0, \alpha _1, \alpha _2} u \Vert \le \frac{2}{p+1} \Vert x_0^{\frac{p+2}{2}} \partial ^{\alpha _0+1, \alpha _1, \alpha _2} u \Vert . \end{aligned}$$

(4)

Now, we establish a useful estimate.

Lemma 2

Let $u \in C^{\infty }_0([0, + \infty [ \times {\mathbb {R}}^2)$ such that $\mathrm{supp} \, u \subseteq [0,h[ \times ]-T, T[^2$. Let $\varphi \in C^{\infty }_0({\mathbb {R}})$ such that $\mathrm{supp} \, \varphi \subseteq {\mathbb {R}} \setminus ]-nT, nT[$, with $n \ge 2$. For every $r \le 0$, $s \in {\mathbb {R}}$ and $p \ge s+r$, it results

$$\begin{aligned} \left\| \varphi A_s u \right\| _{L^2(\varOmega )} \le \frac{c_{p,r,s}}{[(n-1)T]^p} \Vert u \Vert _{H^{0,0,r}(\varOmega )}. \end{aligned}$$

Proof

In order to obtain the claim, we follow analogous techniques used in the proof of Lemma 3.2 in [3]. For the reader’s convenience, we present the demonstration. We have

$$\begin{aligned} (\varphi A_s u)(x)= & {} \frac{1}{2 \pi } \int _{-\infty }^{+ \infty } e^{i x_2 \xi _2} \varphi (x_2) (1 +|\xi _2|^2)^{\frac{s}{2}} {\widehat{u}}(x_0, x_1, \xi _2) d \xi _2 \nonumber \\= & {} \frac{1}{2 \pi } \int \int _{{\mathbb {R}}^2} e^{i (x_2-y_2) \xi _2} \varphi (x_2) (1 +|\xi _2|^2)^{\frac{s}{2}} u(x_0, x_1, y_2) dy_2 d \xi _2 \nonumber \\= & {} \frac{i^m}{2 \pi } \int \int _{{\mathbb {R}}^2} e^{i (x_2-y_2) \xi _2} \frac{\varphi (x_2) u(x_0, x_1, y_2)}{(x_2-y_2)^m} \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} dy_2 d \xi _2 \nonumber \\= & {} \frac{i^m \varphi (x_2)}{2 \pi } \int _{-\infty }^{+ \infty } \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} d\xi _2 \int _{-\infty }^{+ \infty } e^{i (x_2-y_2) \xi _2} \nonumber \\&\quad u(x_0, x_1, y_2) \frac{\psi \left( \frac{x_2-y_2}{(n-1)T} \right) }{(x_2-y_2)^m} dy_2, \end{aligned}$$

(5)

where $m \in {\mathbb {N}}$ and $\psi \in C^{\infty }({\mathbb {R}})$ such that $\psi (\tau )=1$ if $|\tau |\ge 1$, $\psi (\tau )=0$ if $|\tau | \le \frac{1}{2}$.

By using (5), we get

$$\begin{aligned} (\varphi A_s u)(x)= & {} \frac{i^m \varphi (x_2)}{2 \pi } \int _{-\infty }^{+ \infty } \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} \, u(x_0, x_1, x_2)\\&\quad * \left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) d\xi _2, \end{aligned}$$

and also

$$\begin{aligned} {\mathcal {F}}_{x_2}(\varphi A_s u)(x_0,x_1, \eta _2)= & {} \frac{i^m {\widehat{\varphi }}(\eta _2)}{2 \pi } * \int _{-\infty }^{+ \infty } \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} \, {\widehat{u}}(x_0, x_1, \eta _2) \nonumber \\&\cdot {\mathcal {F}}_{x_2} \left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) d\xi _2, \end{aligned}$$

(6)

where

$$\begin{aligned} {\mathcal {F}}_{x_2}\left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) = \int _{-\infty }^{+ \infty } e^{i x_2 (\xi _2- \eta _2)} \psi \left( \frac{x_2}{(n-1)T} \right) \frac{1}{x_2^m} dx_2, \end{aligned}$$

Easily, we deduce

$$\begin{aligned}&(1+(\xi _2-\eta _2)^r) {\mathcal {F}}_{x_2}\left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) \\&\quad = \int _{-\infty }^{+ \infty } e^{i x_2 (\xi _2- \eta _2)} \psi \left( \frac{x_2}{(n-1)T} \right) \frac{1}{x_2^m} dx_2 \\&\quad + i^r \sum _{j=0}^r \left( \begin{array}{c} r \\ j \end{array} \right) \int _{-\infty }^{+ \infty } e^{i x_2 (\xi _2- \eta _2)} \partial ^j_{x_2} \psi \left( \frac{x_2}{(n-1)T} \right) \partial ^{r-j}_{x_2} \frac{1}{x_2^m} dx_2, \end{aligned}$$

and, then,

$$\begin{aligned} \left| {\mathcal {F}}_{x_2}\left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) \right| \le \frac{c_{r,m}}{(1+(\xi _2-\eta _2)^2)^{\frac{r}{2}}} \left( \frac{1}{(n-1)T} \right) ^{m-2}. \end{aligned}$$

(7)

Making use of (6) and (7), we obtain

$$\begin{aligned} \Vert \varphi A_s u \Vert= & {} \Vert {\mathcal {F}}_{x_2}(\varphi A_s u) \Vert \\\le & {} \frac{1}{2 \pi } \Vert {\widehat{\varphi }}\Vert _{L^1({\mathbb {R}})} \left\| \int _{-\infty }^{+ \infty } \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} {\widehat{u}}(x_0, x_1, \eta _2) \right. \\&\quad \left. {\mathcal {F}}_{x_2}\left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) d\xi _2 \right\| _{L^2(\varOmega )}\\\le & {} c \int _{-\infty }^{+ \infty } \left\| \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} {\widehat{u}}(x_0, x_1, \eta _2)\right. \\&\quad \left. {\mathcal {F}}_{x_2} \left( \psi \left( \frac{x_2}{(n-1)T} \right) \frac{e^{i x_2 \xi _2}}{x_2^m} \right) \right\| _{L^2(\varOmega )} d\xi _2\\\le & {} \frac{c_{r,m}}{[(n-1)T]^{m-2}} \int _{-\infty }^{+ \infty } \frac{\left\| \partial ^m_{\xi _2} (1 +|\xi _2|^2)^{\frac{s}{2}} {\widehat{u}}(x_0, x_1, \eta _2) \right\| _{L^2(\varOmega )}}{(1+(\xi _2 - \eta _2)^2)^{\frac{r}{2}}} d\xi _2. \end{aligned}$$

From the previous inequality and the Peetre inequality (see [9], pag. 17), it follows

$$\begin{aligned}&\Vert \varphi A_s u \Vert _{L^2(\varOmega )} \le \frac{c_{r,m,s}}{[(n-1)T]^{m-2}} \nonumber \\&\quad \int _{-\infty }^{+ \infty } \frac{\left\| (1 +|\xi _2|^2)^{\frac{s}{2} - \frac{m+1}{2}} {\widehat{u}}(x_0, x_1, \eta _2) \right\| _{L^2(\varOmega )}}{(1+(\xi _2 - \eta _2)^2)^{\frac{r}{2}}} d\xi _2. \end{aligned}$$

(8)

If $m \ge s+r+2$, setting $p=m-2$ in (8), it results

$$\begin{aligned} \Vert \varphi A_s u \Vert _{L^2(\varOmega )} \le \frac{c_{p,r,s}}{[(n-1)T]^{p}} \Vert u \Vert _{H^{0,0,r}(\varOmega )}, \end{aligned}$$

where $c_{p,r,s}$ is independent of n and T. $\square$

Taking into account Lemma 2, we deduce

Lemma 3

Let $\varphi \in C^{\infty }_0({\mathbb {R}})$ such that $\varphi (\tau )=0$, for $|\tau | \le 1$. For every $\varepsilon >0$, for every $r \le 0$ and $s \in {\mathbb {R}}$ there exists $n>1$ such that

$$\begin{aligned} \left\| \varphi \left( \frac{x_2}{(n-1)T} \right) A_s u \right\| _{L^2(\varOmega )} \le \varepsilon \Vert u \Vert _{H^{0,0,r}(\varOmega )}. \end{aligned}$$

In the following, we establish a priori estimates in $L^2(\varOmega _T)$, where $\varOmega _{h,T}= [0, h[ \times ]-T, T[^2$, for functions belonging to $C_0^{\infty }(\varOmega _{h,T})$.

Theorem 2

For every $h,T>0$, there exists a positive constant c such that

$$\begin{aligned} \Vert \partial _{x_0} u \Vert + \Vert u \Vert \le c \left( \Vert \partial _{x_1} Q u \Vert + \Vert \partial _{x_2} Q u \Vert \right) , \quad \forall u \in C_0^{\infty }({\overline{\varOmega }}): \ \mathrm{supp} \, u \subseteq \varOmega _{h,T}. \end{aligned}$$

(9)

Proof

By means of a translation with respect to $x_2$ in T, we consider the function

$$\begin{aligned} v(x_0,x_1,x_2) = u(x_0,x_1, x_2 -T), \quad \mathrm{in} \ \varOmega _T'= ]0, + \infty [ \times ]-T, T[ \times ]0, 2T[. \end{aligned}$$

We extend the function v in even manner in $]-2T, 2T[$. It results

$$\begin{aligned} v(x_0,x_1, -x_2) = v(x_0, x_1,x_2), \quad \mathrm{in} \ \varOmega _T''= ]0, + \infty [ \times ]-T, T[ \times ]-2T, 2T[. \end{aligned}$$

We consider the following Fourier development of the function v:

$$\begin{aligned} v(x_0,x_1,x_2)= & {} \sum _{n=-\infty }^{+\infty } c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}} \\= & {} \sum _{n=-\infty }^{+\infty } v_n(x_0,x_1, x_2), \end{aligned}$$

where $\omega _0= \frac{2 \pi }{4T} = \frac{\pi }{2 T}$ and

$$\begin{aligned} c_n(x_0,x_1) = \frac{1}{2 \sqrt{T}} \int _{-2T}^{2T} v(x_0,x_1,x_2) e^{- in \omega _0 x_2} dx_2. \end{aligned}$$

We remark that the Fourier coefficients $c_n$ are real. We apply the operator Q to $v_n$ obtaining

$$\begin{aligned} Qv_n(x_0,x_1,x_2)= & {} \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}} \Big [ - \partial _{x_0}^3 c_n(x_0,x_1) + \partial _{x_1}^2 \partial _{x_0} c_n(x_0,x_1) \\&- n^2 \omega _0^2 x_1^2 \partial _{x_0} c_n(x_0,x_1) - i n^3 \omega _0^3 b x_1^3 c_n(x_0,x_1) \Big ] \\= & {} L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}, \end{aligned}$$

where we set

$$\begin{aligned} L_n c_n(x_0,x_1)= & {} - \partial _{x_0}^3 c_n(x_0,x_1) + \partial _{x_1}^2\partial _{x_0} c_n(x_0,x_1) - n^2 \omega _0^2 x_1^2 \partial _{x_0} c_n(x_0,x_1) \\&- i n^3 \omega _0^3 b x_1^3 c_n(x_0,x_1). \end{aligned}$$

It results

$$\begin{aligned} Qv(x_0,x_1,x_2) = \sum _{n=-\infty }^{+\infty } L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

We estimate the Fourier coefficients $c_n(x_0,x_1)$ by means of $L_nc_n(x_0,x_1)$ in $L^2$. To this aim, let us consider the inner products

$$\begin{aligned}&(L_n c_n, x_0 \partial _{x_0}^2 c_n) + (x_0 \partial _{x_0}^2 c_n, L_n c_n) \nonumber \\&\quad = - 2 (\partial _{x_0}^3 c_n, x_0 \partial _{x_0}^2 c_n) + 2 (\partial _{x_0} \partial _{x_1}^2 c_n, x_0 \partial _{x_0}^2 c_n) - 2 n^2 \omega _0^2 (x_1^2 \partial _{x_0} c_n, x_0 \partial _{x_0}^2 c_n) \nonumber \\&\quad = 2 \Vert \partial _{x_0}^2 c_n \Vert ^2 -2 (\partial _{x_1} \partial _{x_0} c_n, x_0 \partial _{x_0}^2 \partial _{x_1} c_n) + 2 n^2 \omega _0^2 \Vert x_1 \partial _{x_0} c_n \Vert ^2 \nonumber \\&\quad = 2 \Vert \partial _{x_0}^2 c_n \Vert ^2 + 2 \Vert \partial _{x_1} \partial _{x_0} c_n \Vert ^2 + 2 n^2 \omega _0^2 \Vert x_1 \partial _{x_0} c_n \Vert ^2. \end{aligned}$$

(10)

From which we have

$$\begin{aligned} \Vert \partial _{x_0}^2 c_n \Vert ^2 + \Vert \partial _{x_1} \partial _{x_0} c_n \Vert ^2 + n^2 \omega _0^2 \Vert x_1 \partial _{x_0} c_n \Vert ^2 \le c \Vert x_0 L_n c_n \Vert ^2. \end{aligned}$$

(11)

Let us evaluate the inner products

$$\begin{aligned} (L_n \partial _{x_1} c_n, x_0 \partial _{x_0}^2 \partial _{x_1} c_n) + (x_0 \partial _{x_0}^2 \partial _{x_1} c_n, L_n \partial _{x_1} c_n). \end{aligned}$$

Proceeding as in (10), we obtain

$$\begin{aligned}&\Vert \partial _{x_0}^2 \partial _{x_1} c_n \Vert ^2 + \Vert \partial _{x_1}^2 \partial _{x_0} c_n \Vert ^2 + n^2 \omega _0^2 \Vert x_1 \partial _{x_0} \partial _{x_1} c_n \Vert ^2 \\&\quad = (L_n \partial _{x_1} c_n, x_0 \partial _{x_0}^2 \partial _{x_1} c_n) + (x_0 \partial _{x_0}^2 \partial _{x_1} c_n, L_n \partial _{x_1} c_n) \\&\quad = (\partial _{x_1} L_n c_n, x_0 \partial _{x_0}^2 \partial _{x_1} c_n) + (x_0 \partial _{x_0}^2 \partial _{x_1} c_n, \partial _{x_1} L_n c_n) \\&\quad + 4 (x_1 n^2 \omega _0^2 \partial _{x_0} c_n, x_0 \partial _{x_0}^2 \partial _{x_1} c_n). \end{aligned}$$

Hence, we deduce

$$\begin{aligned}&\frac{1}{n^2 \omega _0^2} \Vert \partial _{x_0}^2 \partial _{x_1} c_n \Vert ^2 + \frac{1}{n^2 \omega _0^2} \Vert \partial _{x_1}^2 \partial _{x_0} c_n \Vert ^2 + \Vert x_1 \partial _{x_0} \partial _{x_1} c_n \Vert ^2 \\&\quad \le \frac{2}{n^2 \omega _0^2} \Vert x_0 \partial _{x_1} L_n c_n \Vert \Vert \partial _{x_0}^2 \partial _{x_1} c_n \Vert + 4 \Vert x_0 x_1 \partial _{x_0} c_n \Vert \Vert \partial _{x_0}^2 \partial _{x_1} c_n \Vert . \end{aligned}$$

As a consequence, we have

$$\begin{aligned}&\frac{1}{n^2 \omega _0^2} \Vert \partial _{x_0}^2 \partial _{x_1} c_n \Vert ^2 + \frac{1}{n^2 \omega _0^2} \Vert \partial _{x_1}^2 \partial _{x_0} c_n \Vert ^2 + \Vert x_1 \partial _{x_0} \partial _{x_1} c_n \Vert ^2 \nonumber \\&\quad \le \frac{c}{n^2 \omega _0^2} \Vert x_0 \partial _{x_1} L_n c_n \Vert ^2 + c n^2 \omega _0^2 \Vert x_1 \partial _{x_0} c_n \Vert ^2. \end{aligned}$$

(12)

Making use of (11) and (12), we get

$$\begin{aligned}&\Vert \partial _{x_0}^2 c_n \Vert ^2 + \Vert \partial _{x_1} \partial _{x_0} c_n \Vert ^2 + n^2 \omega _0^2 \Vert x_1 \partial _{x_0} c_n \Vert ^2 \nonumber \\&\quad + \frac{1}{n^2 \omega _0^2} \Vert \partial _{x_0}^2 \partial _{x_1} c_n \Vert ^2 + \frac{1}{n^2 \omega _0^2} \Vert \partial _{x_1}^2 \partial _{x_0} c_n \Vert ^2 \nonumber \\&\quad + \Vert x_1 \partial _{x_0} \partial _{x_1} c_n \Vert ^2 \le c \Vert x_0 L_n c_n \Vert ^2 + \frac{c}{n^2 \omega _0^2} \Vert x_0 \partial _{x_1} L_n c_n \Vert ^2 \nonumber \\&\quad \ = \frac{c}{n^2 \omega _0^2} \Vert x_0 \partial _{x_1} L_n c_n \Vert ^2 + \frac{c}{n^2 \omega _0^2} \Vert i x_0 n \omega _0 L_n c_n \Vert ^2. \end{aligned}$$

(13)

Let us consider $v \in C_0^{\infty }(]0, + \infty [ \times ]-T,T[ \times ]0,2T[)$ and we still denote by v its even extension in $]0, + \infty [ \times ]-T,T[ \times ]-2T,2T[$. Let us develop v in Fourier’s series with respect to $x_2$:

$$\begin{aligned} v(x_0,x_1,x_2) = \sum _{n=- \infty }^{+ \infty } c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}, \end{aligned}$$

from which it follows

$$\begin{aligned} Q v(x_0,x_1,x_2) = \sum _{n=- \infty }^{+ \infty } L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}, \end{aligned}$$

Hence, it results

$$\begin{aligned} x_1 \partial _{x_0} \partial _{x_1} v(x_0,x_1,x_2) = \sum _{n=- \infty }^{+ \infty } x_1 \partial _{x_0} \partial _{x_1} c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

Applying the Parseval inequality, we have

$$\begin{aligned} \Vert x_1 \partial _{x_0} \partial _{x_1} v \Vert ^2= & {} \big \Vert \Vert x_1 \partial _{x_0} \partial _{x_1} v \Vert ^2_{]-2T,2T[} \big \Vert ^2_{]0, + \infty [ \times ]-T,T[} \nonumber \\= & {} \left\| \sum _{n=- \infty }^{+ \infty } | x_1 \partial _{x_0} \partial _{x_1} c_n |^2 \right\| ^2_{]0, + \infty [ \times ]-T,T[} \nonumber \\\le & {} \sum _{n=- \infty }^{+ \infty } \Vert x_1 \partial _{x_0} \partial _{x_1} c_n \Vert ^2 . \end{aligned}$$

(14)

Taking into account (13) and (14), we obtain

$$\begin{aligned} \Vert x_1 \partial _{x_0} \partial _{x_1} v \Vert ^2\le & {} \sum _{n=- \infty }^{+ \infty } \Vert x_1 \partial _{x_0} \partial _{x_1} c_n \Vert ^2 \nonumber \\\le & {} \sum _{n=- \infty }^{+ \infty } \left[ \frac{c}{n^2 \omega _0^2} \Vert x_0 \partial _{x_1} L_n c_n \Vert ^2 + \frac{c}{n^2 \omega _0^2} \Vert i x_0 n \omega _0 L_n c_n \Vert ^2 \right] \nonumber \\\le & {} \frac{c}{\omega _0^2} \sum _{n=- \infty }^{+ \infty } \frac{1}{n^2} \left[ \Vert x_0 \partial _{x_1} L_n c_n \Vert ^2 + \Vert i n x_0 \omega _0 L_n c_n \Vert ^2 \right] . \end{aligned}$$

(15)

We remark that

$$\begin{aligned} x_0 \partial _{x_1} Q v(x_0,x_1,x_2) = \sum _{n=- \infty }^{+ \infty } x_0 \partial _{x_1} L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

For the Parseval inequality, it results

$$\begin{aligned} \Vert x_0 \partial _{x_1} Q v \Vert ^2= & {} \big \Vert \Vert x_0 \partial _{x_1} Q v \Vert ^2_{]-2T,2T[} \big \Vert ^2_{]0, + \infty [ \times ]-T,T[} \nonumber \\= & {} \left\| \sum _{n=- \infty }^{+ \infty } | x_0 \partial _{x_1} L_n c_n |^2 \right\| ^2_{]0, + \infty [ \times ]-T,T[} . \end{aligned}$$

(16)

Moreover, we remark that

$$\begin{aligned} x_0 \partial _{x_2} Q v(x_0,x_1,x_2) = \sum _{n=- \infty }^{+ \infty } i n \omega _0 x_0 L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

Applying, again, the Parseval inequality, we have

$$\begin{aligned} \Vert x_0 \partial _{x_2} Q v \Vert ^2= & {} \big \Vert \Vert x_0 \partial _{x_2} Q v \Vert ^2_{]-2T,2T[} \big \Vert ^2_{]0, + \infty [ \times ]-T,T[} \nonumber \\= & {} \left\| \sum _{n=- \infty }^{+ \infty } | i n \omega _0 x_0 L_n c_n |^2 \right\| ^2_{]0, + \infty [ \times ]-T,T[} . \end{aligned}$$

(17)

Making use of (15), (16) and (17), we obtain

$$\begin{aligned} \Vert x_1 \partial _{x_0} \partial _{x_1} v \Vert ^2\le & {} c \sum _{n=- \infty }^{+ \infty } \frac{1}{n^2} \left[ \Vert x_0 \partial _{x_1} L_n c_n \Vert ^2 + \Vert i n \omega _0 L_n c_n \Vert ^2 \right] \nonumber \\\le & {} c \left[ \Vert x_0 \partial _{x_1} Qv \Vert ^2 + \Vert x_0 \partial _{x_2} Qv \Vert ^2 \right] \end{aligned}$$

(18)

On the other hand, it results

$$\begin{aligned} 0= & {} \int _{\varOmega _T''} \partial _{x_1} x_1 \left( \partial _{x_0} v \right) ^2 dx \\= & {} \int _{\varOmega _T''} \left( \partial _{x_0} v \right) ^2 dx + \int _{\varOmega _T''} 2 x_1 \partial _{x_0} v \partial _{x_0} \partial _{x_1} v dx. \end{aligned}$$

From which it follows

$$\begin{aligned} \Vert \partial _{x_0} v \Vert ^2= & {} - 2 \int _{\varOmega _T''} x_1 \left( \partial _{x_0} v \right) \left( \partial _{x_0} \partial _{x_1} v \right) dx \\\le & {} 2 \Vert \partial _{x_0} v \Vert \Vert x_1 \partial _{x_0} \partial _{x_1} v \Vert . \end{aligned}$$

Hence, we have

$$\begin{aligned} \Vert \partial _{x_0} v \Vert ^2 \le 4 \Vert x_1 \partial _{x_0} \partial _{x_1} v \Vert ^2. \end{aligned}$$

(19)

From (18) and (19), we deduce

$$\begin{aligned} \Vert \partial _{x_0} v \Vert ^2 \le c \left( \Vert x_0 \partial _{x_1} Q v \Vert ^2 + \Vert x_0 \partial _{x_2} Q v \Vert ^2 \right) . \end{aligned}$$

By using Lemma 1, it results

$$\begin{aligned} \Vert \partial _{x_0} v \Vert ^2 + \Vert v \Vert ^2 \le c \left( \Vert x_0 \partial _{x_1} Q v \Vert ^2 + \Vert x_0 \partial _{x_2} Q v \Vert ^2 \right) . \end{aligned}$$

(20)

Let us remark

$$\begin{aligned} x_0 \partial _{x_1} Qv= & {} \sum _{n=- \infty }^{+ \infty } x_0 \partial _{x_1} L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}, \\ x_0 \partial _{x_2} Qv= & {} \sum _{n=- \infty }^{+ \infty } i n \omega _0 x_0 L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

As a consequence, we have

$$\begin{aligned} \Vert x_0 \partial _{x_1} Qv \Vert ^2_{]-2T,2T[}= & {} \sum _{n=- \infty }^{+ \infty } | x_0 \partial _{x_1} L_n c_n |^2, \end{aligned}$$

(21)

$$\begin{aligned} \Vert x_0 \partial _{x_2} Qv \Vert ^2_{]-2T,2T[}= & {} \sum _{n=- \infty }^{+ \infty } | i n \omega _0 x_0 L_n c_n|^2. \end{aligned}$$

(22)

Furthermore, we obtain

$$\begin{aligned} ( x_0 \partial _{x_1} Qv) (x_2) + ( x_0 \partial _{x_1} Qv) (- x_2)= & {} \sum _{n=- \infty }^{+ \infty } 2 x_0 \partial _{x_1} L_n c_n \frac{\cos (n \omega _0 x_2)}{2 \sqrt{T}}, \end{aligned}$$

(23)

$$\begin{aligned} ( x_0 \partial _{x_2} Qv) (x_2) + ( x_0 \partial _{x_2} Qv) (- x_2)= & {} \sum _{n=- \infty }^{+ \infty } 2 i n \omega _0 x_0 L_n c_n \frac{\cos (n \omega _0 x_2)}{2 \sqrt{T}}. \end{aligned}$$

(24)

By using (21), (22), (23) and (24), we deduce

$$\begin{aligned}&\Vert x_0 \partial _{x_1} Qv \Vert ^2_{]-2T, 2T[} + \Vert x_0 \partial _{x_2} Qv \Vert ^2_{]-2T, 2T[} \nonumber \\&\quad = \sum _{n=- \infty }^{+ \infty } | x_0 \partial _{x_1} L_n c_n |^2 + \sum _{n=- \infty }^{+ \infty } | i n \omega _0 x_0 L_n c_n |^2 \nonumber \\&\quad = \Vert x_0 (\partial _{x_1} Qv) (x_2) + x_0 (\partial _{x_1} Qv) (- x_2) \Vert ^2_{]-2T,2T[} \nonumber \\&\qquad + \Vert x_0 (\partial _{x_2} Qv) (x_2) + x_0 (\partial _{x_2} Qv) (- x_2) \Vert ^2_{]-2T,2T[} \nonumber \\&\quad = 2 \Vert x_0 (\partial _{x_1} Qv) (x_2) + x_0 (\partial _{x_1} Qv) (- x_2) \Vert ^2_{]0,2T[} \nonumber \\&\qquad + 2 \Vert x_0 (\partial _{x_2} Qv) (x_2) + x_0 (\partial _{x_2} Qv) (- x_2) \Vert ^2_{]0,2T[} \nonumber \\&\quad \le 2 \big ( \Vert x_0( \partial _{x_1} Qv) (x_2) \Vert ^2_{]0,2T[} + \Vert x_0 (\partial _{x_1} Qv) (- x_2) \Vert ^2_{]0,2T[}\nonumber \\&\qquad + \Vert x_0 (\partial _{x_2} Qv) (x_2) \Vert ^2_{]0,2T[} + \Vert x_0 (\partial _{x_2} Qv) (- x_2) \Vert ^2_{]0,2T[} \big ). \end{aligned}$$

(25)

Now, we want to estimate directly the norms. Let us start from

$$\begin{aligned} x_0 \left( \partial _{x_1} Q v \right) (x_2) = \sum _{n=- \infty }^{+\infty } x_0 \partial _{x_1} L_n c_n(x_0,x_1) \frac{e^{in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

It results

$$\begin{aligned}&\left\| x_0 \left( \partial _{x_1} Q v \right) (x_2) \right\| _{]0,2T[}^2 \\&\quad = \frac{1}{4 T} \sum _{n=- \!\infty }^{+\infty } \sum _{h=- \infty }^{+\infty } x_0^2 \partial _{x_1} L_n c_n(x_0,x_1) \left( \partial _{x_1} L_h c_h(x_0,x_1) \right) ^* \!\int _0^{2T} e^{in \omega _0 x_2} e^{- ih \omega _0 x_2} d x_2. \end{aligned}$$

Let us compute the other norm remembering that

$$\begin{aligned} x_0 \left( \partial _{x_1} Q v \right) (- x_2) = \sum _{n=- \infty }^{+\infty } x_0 \partial _{x_1} L_n c_n(x_0,x_1) \frac{e^{- in \omega _0 x_2}}{2 \sqrt{T}}. \end{aligned}$$

We have

$$\begin{aligned}&\left\| x_0 \left( \partial _{x_1} Q v \right) (- x_2) \right\| _{]0,2T[}^2 \\&\quad = \frac{1}{4 T} \!\sum _{n=- \infty }^{+\infty }\! \sum _{h=- \infty }^{+\infty } x_0^2 \partial _{x_1} L_n c_n(x_0,x_1)\! \left( \partial _{x_1} L_h c_h(x_0,x_1) \right) ^* \int _0^{2T} e^{- in \omega _0 x_2} e^{ih \omega _0 x_2} d x_2. \end{aligned}$$

From which, it follows

$$\begin{aligned} \left\| x_0 \left( \partial _{x_1} Q v \right) (x_2) \right\| _{]0,2T[}^2 = \left\| x_0 \left( \partial _{x_1} Q v \right) (- x_2) \right\| _{]0,2T[}^2. \end{aligned}$$

Moreover, making use of (25) and (20), we obtain

$$\begin{aligned} \Vert \partial _{x_0} v \Vert ^2_{\varOmega _T''} + \Vert v \Vert ^2_{\varOmega _T''}\le & {} c \left( \Vert x_0 \partial _{x_1} Q v \Vert ^2_{\varOmega _T''} + \Vert x_0 \partial _{x_2} Q v \Vert ^2_{\varOmega _T''} \right) \\\le & {} 2 c \left( \Vert x_0 \partial _{x_1} Q v \Vert ^2_{\varOmega _T'} + \Vert x_0 \partial _{x_2} Q v \Vert ^2_{\varOmega _T'} \right) . \end{aligned}$$

Since $v(x_0,x_1, x_2) = u(x_0,x_1, x_2 -T)$, for every $(x_0,x_1, x_2) \in \varOmega _T'$, we have

$$\begin{aligned} \Vert \partial _{x_0} u \Vert _{\varOmega _T} + \Vert u \Vert _{\varOmega _T} \le c \left( \Vert x_0 \partial _{x_1} Q u \Vert _{\varOmega _T} + \Vert x_0 \partial _{x_2} Q u \Vert _{\varOmega _T} \right) , \end{aligned}$$

(26)

from which the claim follows. $\square$

Let us remark that the positive constant c in (26) does not depend on T but only on $x_1$. As a consequence, the following result holds:

Corollary 1

For every $h,T>0$, there exists a positive constant c such that

$$\begin{aligned} \Vert \partial _{x_0} u \Vert + \Vert u \Vert \le c \left( \Vert \partial _{x_1} Q u \Vert + \Vert \partial _{x_2} Q u \Vert \right) , \end{aligned}$$

(27)

for every $u \in C_0^{\infty }({\overline{\varOmega }})$ such that $\mathrm{supp} \, u \subseteq [0,T[ \times ]-T,T[ \times ]-nT, nT[$, for every $n \in {\mathbb {N}}$.

4 A priori estimate in Sobolev spaces

In the following, we establish a priori estimate in the Sobolev spaces.

Theorem 3

For every $s>0$, it results

$$\begin{aligned} \Vert \partial _{x_0} u \Vert _{H^{0,0,-s}} \!+\! \Vert u \Vert _{H^{0,0,-s}} \!\le \! c \left( \Vert Q u \Vert _{H^{0,1,-s}} + \Vert Q u \Vert _{H^{0,0,-s+1}}\right) , \quad \forall u \in C^{\infty }_0(\varOmega _{h,T}),\nonumber \\ \end{aligned}$$

(28)

where $\varOmega _{h,T} = [0,h[ \times ]-T,T[^2$.

Proof

Let $\varphi \in C_0^{\infty }({\mathbb {R}})$ such that $\varphi (x)=1$ in $[-(n-1) T, (n-1) T]$ and $\mathrm{supp} \, \varphi \subseteq ]-nT, nT[$, with $n>1$. For every $u \in C^{\infty }_0(\varOmega _{h,T})$, we set $v_s= \varphi A_s u$, where $A_s$ is the pseudodifferential operator defined as:

$$\begin{aligned} A_s u = \frac{1}{2 \pi } \int _{{\mathbb {R}}} e^{i x_2 \xi _2} (1+ |\xi _2|^2)^{-\frac{s}{2}} {\widehat{u}}(x_0,x_1,\xi _2) d \xi _2, \end{aligned}$$

with $s>0$. Applying (27) to $v_s$, we have

$$\begin{aligned} \Vert \partial _{x_0} v_s \Vert + \Vert v_s \Vert\le & {} c \left( \Vert \partial _{x_1} Qv_s \Vert + \Vert \partial _{x_2} Qv_s \Vert \right) \nonumber \\= & {} c \left( \Vert \partial _{x_1} Q \varphi A_s u \Vert + \Vert \partial _{x_2} Q \varphi A_s u \Vert \right) \nonumber \\\le & {} c \left( \Vert \varphi \partial _{x_1} Q A_s u \Vert + \Vert \partial _{x_1} [Q,\varphi ] A_s u \Vert \right) \nonumber \\&+ c \left( \Vert \partial _{x_2} \varphi Q A_s u \Vert + \Vert \partial _{x_2} [Q, \varphi ] A_s u \Vert \right) \nonumber \\\le & {} c \left( \Vert \varphi \partial _{x_1} A_s Q u \Vert + \Vert \partial _{x_1} [Q,\varphi ] A_s u \Vert \right) \nonumber \\&+ c \left( \Vert \varphi \partial _{x_2} A_s Q u \Vert + \Vert [\partial _{x_2}, \varphi ] Q A_s u \Vert + \Vert \partial _{x_2} [Q, \varphi ] A_s u \Vert \right) \nonumber \\= & {} c \Big ( \Vert \varphi A_s \partial _{x_1} Q u \Vert + \Vert \varphi A_s \partial _{x_2} Q u \Vert + \Vert R_1 Qu \Vert \nonumber \\&+ \Vert R_2 u \Vert + \Vert R_3 \partial _{x_0} u \Vert \Big ) +c \Vert [Q,\varphi ] A_s \partial _{x_1} u \Vert , \end{aligned}$$

(29)

where $R_1$, $R_2$ and $R_3$ are regularizing operators with respect to the variable $x_2$ of type

$$\begin{aligned} R_i = \psi \left( \frac{x_2}{(n-1)T} \right) A_s, \quad i=1,2,3, \end{aligned}$$

(30)

with $\psi \in C^{\infty }_0({\mathbb {R}})$ such that $\psi =0$ in $[-1, 1]$, as in Lemma 3, and having used $\partial _{x_1} Q A_s u= A_s \partial _{x_1} Qu$ and $\partial _{x_2} Q A_s u= A_s \partial _{x_2} Qu$.

Making use of Lemmas 1, 3 and (29), we deduce

$$\begin{aligned} \Vert \partial _{x_0} u \Vert _{H^{0,0,-s}} + \Vert u \Vert _{H^{0,0,-s}}\le & {} c \left( \Vert Qu \Vert _{H^{0,1,-s}} + \Vert Qu \Vert _{H^{0,0,-s+1}} + \Vert Qu \Vert _{H^{0,0,-s}} \right) \nonumber \\&+c \left( \Vert R_4 \partial _{x_1} u \Vert + \Vert R_5 \partial _{x_1} \partial _{x_0} u \Vert \right) , \end{aligned}$$

(31)

where $R_4$ and $R_4$ are regularizing operators with respect to the variable $x_2$ of type (30).

Now, written the operator Q as:

$$\begin{aligned} Qu = L(\partial _{x_0} u) + x_1^2 \partial _{x_2}^2 \partial _{x_0} u + b x_1^3 \partial _{x_2}^3 u, \end{aligned}$$

where L is the wave operator, namely $L= \partial _{x_0}^2 + \partial _{x_1}^2$, it results

$$\begin{aligned} (L(\partial _{x_0} u), \partial _{x_0}^2 u) = (Qu, \partial _{x_0}^2 u) - (x_1^2 \partial _{x_2}^2 \partial _{x_0} u, \partial _{x_0}^2 u) - (b x_1^3 \partial _{x_2}^3u, \partial _{x_0}^2 u). \end{aligned}$$

Integrating by parts, we have easily:

$$\begin{aligned} \Vert \partial _{x_0}^2 u \Vert + \Vert \partial _{x_1} \partial _{x_0} u \Vert \le c \left( \Vert \partial _{x_2}^2 \partial _{x_0} u \Vert + \Vert \partial _{x_2}^3 u \Vert + \Vert Q u \Vert \right) . \end{aligned}$$

Making use of Lemma 1, it follows

$$\begin{aligned} \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert \partial _{x_0}^2 u \Vert + \Vert \partial _{x_1} \partial _{x_0} u \Vert \le c \left( \Vert \partial _{x_2}^2 \partial _{x_0} u \Vert + \Vert \partial _{x_2}^3 u \Vert + \Vert Q u \Vert \right) .\nonumber \\ \end{aligned}$$

(32)

Taking into account (31), (32) and Lemma 3, we deduce

$$\begin{aligned} \Vert \partial _{x_0} u \Vert _{H^{0,0,-s}} + \Vert u \Vert _{H^{0,0,-s}}\le & {} c \left( \Vert Q u \Vert _{H^{0,1,-s}} + \Vert Q u \Vert _{H^{0,0,-s+1}} + \Vert Q u \Vert _{H^{0,0,-s}} \right) \\&+ c \left( \Vert \partial _{x_1} u \Vert _{H^{0,0,-s-3}} + \Vert \partial _{x_1} \partial _{x_0} u \Vert _{H^{0,0,-s-3}}\right) \\\le & {} c \left( \Vert Q u \Vert _{H^{0,1,-s}} + \Vert Q u \Vert _{H^{0,0,-s+1}} \right) \\&+ c \left( \Vert u \Vert _{H^{0,0,-s}} + \Vert \partial _{x_0} u \Vert _{H^{0,0,-s}} + \Vert Q u \Vert _{H^{0,0,-s}}\right) . \end{aligned}$$

From which we have

$$\begin{aligned} \Vert \partial _{x_0} u \Vert _{H^{0,0,-s}} + \Vert u \Vert _{H^{0,0,-s}} \le c \left( \Vert Q u \Vert _{H^{0,1,-s}} + \Vert Q u \Vert _{H^{0,0,-s+1}}\right) , \end{aligned}$$

namely (28). $\square$

5 Proof of Theorem 1

For every $u \in C^{\infty }_0( \varOmega _{h,T})$, where $\varOmega _{h,T} =[0,h[ \times ]-T,T[^2$, let $\psi = \, ^t Qu = Qu$ and let $F(\psi )=(f,u)$. It results

$$\begin{aligned} |F( \psi )| \le \Vert f \Vert _{H^{0,0,s}(\varOmega _{h,T})} \Vert u \Vert _{H^{0,0,s}(\varOmega _{h,T})}. \end{aligned}$$

Making use of (28), it follows

$$\begin{aligned} |F( \psi )|\le & {} c \Vert f \Vert _{H^{0,0,s}(\varOmega _{h,T})} \left( \Vert \, ^t Qu \Vert _{H^{0,1,-s}(\varOmega _{h,T})} + \Vert \, ^t Qu \Vert _{H^{0,0,-s+1}(\varOmega _{h,T})} \right) \\\le & {} c' \Vert \psi \Vert _{H^{0,1,-s+1}(\varOmega _{h,T})}. \end{aligned}$$

Hence, the functional F can be extended in $H^{0,1,-s+1}(\varOmega _{h,T})$ and, therefore, there exists $w \in H^{0,-1,s-1}(\varOmega _{h,T})$ such that

$$\begin{aligned} F(\psi ) = (w, \psi ) = (w, \, ^t Qu) = (g,u), \quad \forall u \in C^{\infty }_0(\varOmega _{h,T}). \end{aligned}$$

Then, we have

$$\begin{aligned} Qw=g, \quad \mathrm{in} \ {\mathcal {D}}'(\varOmega _{h,T}). \end{aligned}$$

Written $Qw= L( \partial _{x_0}w) + x_1^2 \partial _{x_2}^2 \partial _{x_0} w+ b x_1^3 \partial _{x_2}^3 w$, we obtain

$$\begin{aligned} L(\partial _{x_0} w) = g - x_1^2 \partial _{x_2}^2 \partial _{x_0} w - b x_1^3 \partial _{x_2}^3 w. \end{aligned}$$

For $s >4$, we deduce $\partial _{x_0} w \in H^{0,0,s-1}$ and, hence, $u \in H^{1,s-1}$. Repeating the same procedure more times, we have that if $g \in H^r$ then $w \in H^{r-2}$. Therefore, if $r \ge 5$, we have

$$\begin{aligned} (w, \, ^tQu) = (g,u), \quad \forall u \in C^{\infty }_0(\varOmega _{h,T}). \end{aligned}$$

(33)

Choosen a suitable u, for instance, such that $u(0,x')=0$, $\partial _{x_0} u(0,x')=0$ and $\partial _{x_0}^2 u(0,x')= \varphi (x')$, with $\varphi \in C^{\infty }_0(]-T,T[^2)$, integrating by parts in the left-hand side of (33), we obtain

$$\begin{aligned} (Qw,u) + \int _{[-T,T]^2} \varphi (x') w dx' = (g,u). \end{aligned}$$

As a consequence, we get

$$\begin{aligned} \int _{[-T,T]^2} \varphi (x') w dx' = 0. \end{aligned}$$

For the arbitrariness of $\varphi$, it follows

$$\begin{aligned} w(0,x') = 0. \end{aligned}$$

Instead, choosing $u \in C^{\infty }_0(\varOmega _{h,T})$ such that $u(0,x')=0$, $\partial _{x_0} u(0,x')=\varphi (x')$ and $\partial _{x_0}^2 u(0,x')= 0$, with $\varphi \in C^{\infty }_0(]-T,T[^2)$, and proceeding as above, it results

$$\begin{aligned} \partial _{x_0} w(0,x') = 0. \end{aligned}$$

Finally, if we chose $u \in C^{\infty }_0(\varOmega _{h,T})$ such that $u(0,x')=\varphi (x')$, $\partial _{x_0} u(0,x')=0$ and $\partial _{x_0}^2 u(0,x')= 0$, with $\varphi \in C^{\infty }_0(]-T,T[^2)$, we obtain

$$\begin{aligned} \partial _{x_0}^2 w(0,x') = 0. \end{aligned}$$

Then we have proved that there exists $w \in H^{r-2}(\varOmega _{h,T})$, with $r \ge 5$, such that

$$\begin{aligned} (w, \, ^tQu) = (g,u), \quad \forall u \in C^{\infty }_0(\varOmega _{h,T}), \end{aligned}$$

$w(0,x')=0$, $\partial _{x_0} w(0,x')=0$ and $\partial _{x_0}^2 w(0,x')= 0$. Hence, if $g \in H^r$, with $r \ge 5$, there exists a solution $w \in H^{r-2}$ to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Qw=g, \quad \mathrm{in} \ \varOmega _{h,T} \\ w(0,x')=0, \ \partial _{x_0} w(0,x')=0, \ \partial _{x_0}^2 w(0,x')= 0 \end{array}\right. } \end{aligned}$$

where $g=if$, with $f \in H^r_{loc}(\varOmega )$. Therefore there exists a solution to problem (1) also in $\varOmega _{h,T}$.

6 Conclusions

The paper deals with a class of hyperbolic operators with triple characteristics. A priori estimate in Sobolev spaces with negative indexes are obtained. Thanks to this estimate, the existence of solutions to the associated Cauchy problem can be established.