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.
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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
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
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
with \(f \in H^r(\varOmega )\), in the Sobolev spaces, obtaining an existence result for solutions.
Let us set
It results
As a consequence, problem (1) becomes
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
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
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:
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
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
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
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
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
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
and also
where
Easily, we deduce
and, then,
Making use of (6) and (7), we obtain
From the previous inequality and the Peetre inequality (see [9], pag. 17), it follows
If \(m \ge s+r+2\), setting \(p=m-2\) in (8), it results
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
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
Proof
By means of a translation with respect to \(x_2\) in T, we consider the function
We extend the function v in even manner in \(]-2T, 2T[\). It results
We consider the following Fourier development of the function v:
where \(\omega _0= \frac{2 \pi }{4T} = \frac{\pi }{2 T}\) and
We remark that the Fourier coefficients \(c_n\) are real. We apply the operator Q to \(v_n\) obtaining
where we set
It results
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
From which we have
Let us evaluate the inner products
Proceeding as in (10), we obtain
Hence, we deduce
As a consequence, we have
Making use of (11) and (12), we get
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 \(x_2\):
from which it follows
Hence, it results
Applying the Parseval inequality, we have
Taking into account (13) and (14), we obtain
We remark that
For the Parseval inequality, it results
Moreover, we remark that
Applying, again, the Parseval inequality, we have
Making use of (15), (16) and (17), we obtain
On the other hand, it results
From which it follows
Hence, we have
By using Lemma 1, it results
Let us remark
As a consequence, we have
Furthermore, we obtain
By using (21), (22), (23) and (24), we deduce
Now, we want to estimate directly the norms. Let us start from
It results
Let us compute the other norm remembering that
We have
From which, it follows
Moreover, making use of (25) and (20), we obtain
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
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
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
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:
with \(s>0\). Applying (27) to \(v_s\), we have
where \(R_1\), \(R_2\) and \(R_3\) are regularizing operators with respect to the variable \(x_2\) of type
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
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:
where L is the wave operator, namely \(L= \partial _{x_0}^2 + \partial _{x_1}^2\), it results
Integrating by parts, we have easily:
Making use of Lemma 1, it follows
Taking into account (31), (32) and Lemma 3, we deduce
From which we have
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
Making use of (28), it follows
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
Then, we have
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
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
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
As a consequence, we get
For the arbitrariness of \(\varphi \), it follows
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
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
Then we have proved that there exists \(w \in H^{r-2}(\varOmega _{h,T})\), with \(r \ge 5\), such that
\(w(0,x')=0\), \(\partial _{x_0} w(0,x')=0\) 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
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.
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Barbagallo, A., Esposito, V. On the Cauchy problem for a class of hyperbolic operators with triple characteristics. Ricerche mat 71, 459–475 (2022). https://doi.org/10.1007/s11587-020-00540-6
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DOI: https://doi.org/10.1007/s11587-020-00540-6