On hyperbolic equations with double characteristics in the presence of transition

The spectrum of \(F(\rho)\), which we denote by \(\operatorname{Spec}(F(\rho))\), has a remarkable importance for the study of the well-posedness of the Cauchy-Dirichlet problem for P.

The Cauchy problem for hyperbolic operators with double characteristics has been widely studied by many authors either in the case in which \(F_{p}(\rho)\) has two real nonzero eigenvalues \(\forall \rho \in\Sigma\) or in the case in which all the nonzero eigenvalues of \(F_{p}(\rho)\) are purely imaginary numbers, \(\forall\rho\in\Sigma\) (see for instance [1, 3–9]). Recently, another class of hyperbolic second order operators with double characteristics has been considered in [2]. For this class the \(C^{\infty}\) well-posedness of the Cauchy problem is studied. Moreover, Carleman estimates are obtained for non-effectively hyperbolic operators. In [10], for a different class of hyperbolic second order operators some energy estimates are established and the \(C^{\infty}\) well-posedness of the Cauchy problem for non-effectively hyperbolic operators is studied. We emphasize that in [2] and [10] the authors obtain a priori estimates when \(\Sigma= \Sigma_{-} \sqcup \Sigma_{0}\). Instead we get a priori estimates when \(\Sigma= \Sigma_{-} \sqcup\Sigma_{0} \sqcup\Sigma_{+}\) or \(\Sigma= \Sigma_{-} \sqcup \Sigma_{0}\) or \(\Sigma= \Sigma_{0} \sqcup\Sigma_{+}\) or \(\Sigma= \Sigma_{-}\) or \(\Sigma= \Sigma_{+}\). In fact, in the class of operators (1), studied also in [11, 12] and [13], both in the case in which \(F_{p}(\rho)\) has two distinct real eigenvalues and in the case in which all the eigenvalues are purely imaginary numbers can occur. Namely, on the variety characteristic a transition from a case to another one can be considered. More precisely, if \(p(x, \xi)= \xi_{0}^{2} - \sum_{j=1}^{m} \xi^{2}_{j} +(x_{0}+\lambda-\alpha(x'))^{2} \sum_{j=m+1}^{n} \xi^{2}_{j}\), setting \(\beta(x)= x_{0}+ \lambda-\alpha(x')\), if \(|\nabla_{x'} \alpha(x')|<1\) and \(\beta(x)=0\) (\(\xi_{0}=\xi_{1}=\cdots=\xi_{m}=0\), \(\sum_{j=m+1}^{n} \xi^{2}_{j}=1\)), then \(F_{p}(\rho)\) has two distinct nonzero real eigenvalues. As a consequence, P is effectively hyperbolic. Instead if \(|\nabla_{x'} \alpha(x')| > 1\) and \(\beta(x)=0\) (\(\xi_{0}=\xi_{1}=\cdots=\xi_{m}=0\), \(\sum_{j=m+1}^{n} \xi^{2}_{j}=1\)), \(F_{p}(\rho)\) has two nonzero imaginary eigenvalues, then P is non-effectively hyperbolic. Therefore \(\Sigma_{+}\) is the set of points of Σ for which \(|\nabla_{x'} \alpha(x)|<1\), \(\Sigma_{-}\) is the set of points of Σ for which \(|\nabla_{x'} \alpha(x)|>1\) and \(\Sigma_{0}\) is the set of points of Σ in which \(|\nabla_{x'} \alpha(x)|=1\). Hence, even if we consider the particular class of operators (1), we have a transition from effectively hyperbolic to non-effectively hyperbolic.

It is worth remarking that in the study of hyperbolic operators considered in this note, the major difficulties in order to establish a priori estimates regard to the case in which the function \(\beta(x, \lambda)= x_{0}+ \lambda- \alpha(x')\) assumes positive and negative values in Ω̅. Let us observe that if \(m=1\), setting \(A(x',x'')=(a(x',x''))\), as a result \(C(x',x'')= \operatorname{div}_{x'} a(x',x'') g(x')\). Moreover, if \(a(x',x'')\) is a constant function, then \(C(x',x'')=0\). Therefore, if \(m=1\) and \(A(x',x'')\) is a constant function, assumption (iv) naturally occurs.

The paper is organized as follows. In Section 2 some preliminary notations are given. In Section 3 a priori estimates are proved. In Section 4, estimates in Sobolev spaces with \(s<0\) by means of the pseudodifferential operator theory are obtained. Section 5 deals with a local existence theorem near the boundary. Then a regularity result for the solution u to the Cauchy problem (2) is shown. At last a global existence result is proved in Section 6.

Let \(\alpha=(\alpha_{0}, \alpha', \alpha'') \in\mathbb{N}^{n+1}_{0}\). We denote by \(\partial^{\alpha}\) the derivative of order \(|\alpha|\), while \(\partial^{h}_{x_{j}}\) means, as usually, the derivative of order h with respect to \(x_{j}\) and \(\partial^{h}_{x_{j}, x_{p}}\) denotes the derivative of order h with respect to \(x_{j}\) and \(x_{p}\).

Let us denote by \((\cdot, \cdot)\), \(\Vert \cdot \Vert \), \(\Vert \cdot \Vert _{H^{r}}\) (\(r \in\mathbb{N}_{0}\)) the \(L^{2}\)-scalar product, the \(L^{2}\)-norm and the \(H^{r}\)-norm, respectively.

\(C_{0}^{\infty}(\overline{\Omega})\) is the space of the restrictions to Ω̅ of functions φ belonging to \(C^{\infty}_{0}(\mathbb{R}^{n+1})\) such that φ vanishes with all the derivatives in \([0, + \infty[\, \times\partial\Omega_{0}\).

Now, we are able to prove the following a priori estimate.

As a consequence, we have the following corollary.

Let us, first, prove some preliminary results.

Taking into account Lemma 4.1, it is easy to show the following.

Furthermore, we are able to prove the following.

Next, we prove a result concerning estimates near the boundary.

As a consequence, we establish the following result.

Now, we are able to prove the following theorem.

Now, we prove an estimate in Sobolev spaces with \(s<0\).

Let \(\Omega_{k}=[0, k[\, \times\Omega_{0}\), with \(k>0\); the following local existence theorem near the boundary holds.