Abstract
Symmetrizers for hyperbolic operators are obtained by diagonalizing the Bézoutian matrix of the principal symbols and its derivatives. Such diagonal symmetrizers are applied to the Cauchy problem for hyperbolic operators with triple characteristics. In particular, the Ivrii’s conjecture concerned with strongly hyperbolic operators with triple effectively hyperbolic characteristics is proved for differential operators with time dependent coefficients, also for third order differential operators with two independent variables with analytic coefficients.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
This paper is devoted to the Cauchy problem
where \(t\ge 0\), \(x\in {\mathbb {R}}^n\) and the coefficients \(a_{j,\alpha }(t,x)\) are real valued \(C^{\infty }\) functions in a neighborhood of the origin of \({\mathbb {R}}^{1+n}\) and \(D_x=(D_{x_1},\ldots ,D_{x_n})\), \(D_{x_j}=(1/i)(\partial /\partial x_j)\) and \(D_t=(1/i)(\partial /\partial t)\). The problem is \(C^{\infty }\) well-posed near the origin for \(t\ge 0\) if one can find a \(\delta >0\) and a neighborhood U of the origin of \({\mathbb {R}}^n\) such that (1.1) has a unique solution \(u\in C^{\infty }([0,\delta )\times U)\) for any \(u_j(x)\in C^{\infty }({\mathbb {R}}^n)\). We assume that the principal symbol p is hyperbolic for \(t\ge 0\), that is
has only real roots in \(\tau \) for \((t,x)\in [0,\delta ')\times U'\) and \(\xi \in {\mathbb {R}}^n\) with some \(\delta '>0\) and a neighborhood \(U'\) of the origin which is necessary in order that the Cauchy problem (1.1) is \(C^{\infty }\) well-posed near the origin for \(t\ge 0\) ([17, 20]).
In this paper we are mainly concerned with the case that the multiplicity of the characteristic roots is at most 3. This implies that it is essential to study operators P of the form
which is differential operator in t with coefficients \(a_{j}\in S^0\), classical pseudodifferential operator of order 0, where \(\langle {D}\rangle = \mathrm{Op}((1 + |\xi |^2)^{1/2})\). One can assume that \(a_1(t,x,D)=0\) without loss of generality and hence the principal symbol has the form
With \(U={^t}(D_t^2u,\langle {D}\rangle D_tu,\langle {D}\rangle ^2u)\) the equation \(P u=f\) is reduced to
where \(A, B\in S^0\), \(F={^t}(f,0,0)\) and
Let S be the Bézoutiant of p and \(\partial p/\partial \tau \), that is
then S is nonnegative definite and symmetrizes A, that is SA is symmetric which is easily examined directly, though this is a special case of a general fact (see [15, 28]). Then one of the most important works would be to obtain lower bound of \((\mathrm{Op}(S)U,U)\). The sharp Gårding inequality ([8, 18]) gives a lower bound
which is, in general, too weak to study the Cauchy problem for general weakly hyperbolic operator P, in particular the well posed Cauchy problem with loss of derivatives, although applying this symmetrizer many interesting results are obtained by several authors, see for example [1, 5, 6, 16, 19, 29]. In these works one of the main points is how one can derive a suitable lower bound of \(\mathrm{Op}(S)\) from the hyperbolicity condition assumed on p, that is
In this paper we employ a new idea which is to diagonalize S by an orthogonal matrix T so that \(T^{-1}ST=\Lambda =\mathrm{diag}\,(\lambda _1,\lambda _2,\lambda _3)\) where \(0\le \lambda _1\le \lambda _2\le \lambda _3\) are the eigenvalues of S and reduce the equation to that of \(V=T^{-1}U\); roughly
where \(\Lambda \) symmetrizes \(A^T\). For general nonnegative definite symmetric S it seems that we have nothing new but our S is a special one which is the Bézoutiant of hyperbolic polynomial p and \(\partial p/\partial \tau \). Indeed, as we will see in Sect. 2, one has
Since (1.6) is a symmetrizable system with a diagonal symmetrizer \(\Lambda \), a natural energy will be
and it could be expected that scalar operators \(\mathrm{Op}(\lambda _j)\) reflect the hyperbolicity condition (1.5) quite directly.
If \(p=0\) has a triple characteristic root \(\tau =0\) at \((0,x,\xi )\) such that \((0,x,\tau ,\xi )\) is effectively hyperbolic (see Sect. 4, also [7, 9]) then one sees that \(\partial _ta(0,x,\xi )>0\) and hence \(\partial _t^3\Delta (0,x,\xi )>0\), which follows from (1.5), so that essentially a and \(\Delta \) are polynomials in t of degree 1 and 3 respectively. Then we see that \(\Delta /a\) behaves like a second order polynomial in t which is nonnegative for \(t\ge 0\). Finding a finite number of functions \(\phi _j\) satisfying \(\partial _t\phi _j>0\) and \(\phi _j^2\preceq \Delta /a\), \(\phi _j\big |\partial _t\Delta \big |\preceq \Delta \), \(\phi _j\big |\partial _ta\big |\preceq a\) we estimate the weighted energy \({\mathsf {Re}}\,\big (\mathrm{Op}(\phi _j^{N_j}\Lambda )U,U\big )\) with a suitable \(N_j\in {\mathbb {R}}\). In Sect. 4 this procedure is carried out for operators of order m with effectively hyperbolic critical points with time dependent coefficients, and it is proved that the Cauchy problem is \(C^{\infty }\) well-posed for any lower order term. In Sect. 5 the same assertion is proved for third order operators with two independent variables with analytic coefficients, so that Ivrii’s conjecture is proved for these operators (Theorems 4.1 and 5.1).
In Sect. 3, admitting the existence of such a weight function \(\phi _j\), we explain how to derive energy estimates. To do so we need to estimate the derivatives of \(\Lambda \) and \(A^T\), essentially those of \(\lambda _j\), which is done in Sect. 2.
In the last section we show that the same idea is applicable to hyperbolic operators with general triple characteristics, utilizing a homogeneous third order operator with two independent variables (Theorem 6.1).
2 Daiagonal symmetrizers
Consider
where a(t, X) and b(t, X) are real valued and \(C^{\infty }\) in \((t,X)\in (-c,T)\times W\) with bounded derivatives of all order where W is an open set in \( {\mathbb {R}}^l\) such that \({\bar{X}}\in W\) and \(c>0\) is some positive constant. Assume
that is, \(p(\tau ,t,X)=0\) has only real roots for \((t,X)\in [0,T)\times W\) and has a triple root \(\tau =0\) at \((0,{\bar{X}})\). Moreover assume that there is no triple root in \(t>0\);
Denote
then S is nonnegative definite and S(t, X)A(t, X) is symmetric. Let
be the eigenvalues of S(t, X).
2.1 Behavior of eigenvalues
We show
Proposition 2.1
There exist a neighborhood \({\mathcal U}\) of \((0,\bar{X})\) and \(K>0\) such that
for \((t,X)\in {\mathcal U}\cap \{t>0\}\).
Corollary 2.1
There exists a neighborhood \({\mathcal U}\) of \((0,\bar{X})\) such that
Proof
Recalling \(a(0,\bar{X})=0\) from Proposition 2.1 one can choose \({\mathcal U}\) such that
then the assertion follows immediately from the Implicit function theorem. \(\square \)
Remark 2.1
It may happen \(\Delta (t,X)=0\) for \(t>0\) so that \(p(\tau ,t,X)=0\) has a double root \(\tau \) at (t, X) while \(\lambda _i(t,X)\) are smooth there.
Proof of Proposition 2.1:
Denote \(q(\lambda )=\mathrm{det}\,(\lambda I-S)\);
Let \(\mu _1\le \mu _2\) be the roots of \(q_{\lambda }=\partial q/\partial \lambda =0\) and hence
It is easy to see \(\mu _1=a(1+O(a))\) and \( \mu _2=2+O(a)\) which gives
In the \((\lambda ,\eta )\) plane, the tangent line of the curve \(\eta =q(\lambda )\) at (0, q(0)) intersects with \(\lambda \) axis at \((\Delta /q_{\lambda }(0),0)\) and hence
Since \(q_{\lambda }(0)\le 6a+2a^2+2a^3\) the left inequality of (2.4) is obvious. Compute \(q(\delta a^2)\) with \(\delta >0\). Since \(2a^3-9b^2=\Delta /2+9b^2/2\ge 0\) one has
Here we take \(\delta =2/3+Ka\) then noting \(a(0,\bar{X})=0\) one can choose a neighborhood \({\mathcal U}\) of \((0,\bar{X})\) such that
for \((t,X)\in {\mathcal U}\cap \{t>0\}\). This proves that \(\lambda _1\le \delta a^2\) and hence the right inequality of (2.4). Turn to \(\lambda _2\). Consider \(q(\delta a)\) with \(\delta >0\) again. Note
and choose \(\delta =2-Ka\) which gives
Therefore for any \(K>0\) one can find \({\mathcal U}\) such that \(q(\delta a)>0\) in \({\mathcal U}\cap \{t>0\}\). Since one can assume \(\delta a<\lambda _3\) by (2.8) then \(\delta a\in (\lambda _1,\lambda _2)\) which proves the left inequality of (2.5). Repeating similar arguments one gets
because \(27b^2\le 4 a^3\) in \(t\ge 0\). Taking \(\delta =2+Ka\) one has
Fixing any \(K>4/3\) one can find \({\mathcal U}\) such that \(q(\delta a)<0\) in \({\mathcal U}\cap \{t>0\}\). Since \(\lambda _1<\delta a\) thanks to (2.4) one concludes \((2+Ka)a\in (\lambda _2,\lambda _3)\) which shows the right inequality of (2.5). Finally we check (2.6). It is easy to see that
in \({\mathcal U}\cap \{t>0\}\) if \({\mathcal U}\) is small so that \(3\le \lambda _3\) in \({\mathcal U}\cap \{t>0\}\). Note that
where we take \(\delta =3+Ka^2\) so that
Thus fixing any \(K>1/3\) one can find \({\mathcal U}\) such that \(q(3+Ka^2)>0\) in \({\mathcal U}\cap \{t>0\}\). Since \(3+Ka^2>\lambda _2\) which proves the right inequality of (2.6). \(\square \)
2.2 Behavior of eigenvectors
If we write \(n_{ij}\) for the (i, j)-cofactor of \(\lambda _kI-S\) then \(^t(n_{j1},n_{j2},n_{j3})\) is, if non-trivial, an eigenvector corresponding to \(\lambda _k\). We take \(k=1\), \(j=3\) and hence
is an eigenvector corresponding to \(\lambda _1\) and therefore
is a normalized eigenvector corresponding to \(\lambda _1\). Thanks to Proposition 2.1 and \(b=O(a^{3/2})\) it is clear that there is \(C>0\) such that
Similarly choosing \(k=2, j=2\) and \(k=3, j=1\)
are eigenvectors corresponding to \(\lambda _2\) and \(\lambda _3\) respectively and
are normalized eigenvectors corresponding to \(\lambda _j\), \(j=2,3\). Thanks to Proposition 2.1 there is \(C>0\) such that
Denote \(T=(\mathbf{t}_1,\mathbf{t}_2, \mathbf{t}_3)=(t_{ij})\) then T is an orthogonal matrix, \({^t}TT=I\), smooth in \((t,X)\in {\mathcal U}\cap \{t>0\}\) which diagonalizes S;
Note that \(\Lambda \) symmetrizes \(A^T=T^{-1}AT\);
Denote \(A^T=({\tilde{a}}_{ij})\). Since \(\Lambda A^T\) is symmetric \({\tilde{a}}_{ij}\) satisfies
which shows, in particular, that \({\tilde{a}}_{21}=O(a){\tilde{a}}_{21}\), \({\tilde{a}}_{31}=O(a^2){\tilde{a}}_{13}\) and \({\tilde{a}}_{32}=O(a){\tilde{a}}_{23}\). Finally in view of (2.9), (2.10) and Proposition 2.1 it is easy to check that
near \((t,X)=(0,{\bar{X}})\).
2.3 Smoothness of eigenvalues
First recall [29, Lemma 3.2]
Lemma 2.1
Assume (2.1). Then
for \(|\alpha |=1\) and \((t,X)\in (0,T)\times W\).
We show
Lemma 2.2
For \(|\alpha |=1\) one has
Proof
Since
it follows from Lemma 2.1 that
From \( q_{\lambda }(\lambda _j)\partial _X^{\alpha }\lambda _j+\partial _X^{\alpha }q(\lambda _j)=0\) one has
Noting \(q_{\lambda }(\lambda _j)=\prod _{k\ne j}(\lambda _j-\lambda _k)\) one sees
thanks to Proposition 2.1 and hence the assertion. \(\square \)
Next estimate \(\partial _t\lambda _j\).
Lemma 2.3
Assuming (2.1) one has
Proof
Repeating the same arguments in the proof of Lemma 2.2 one has
which proves the assertion. \(\square \)
3 How to apply diagonal symmetrizers
Taking
with one space variable \(x\in {\mathbb {R}}\), we explain how to apply diagonal symmetrizers constructed in preceding sections. Assume that
and \(a(0,0)=0\) such that \(p(\tau ,0,0,1)=0\) has the triple root \(\tau =0\) where W is an open interval containing the origin. In what follows we work in a region where \(a(t,x)>0\). With \(U=(\partial _t^2u,\partial _x\partial _tu,\partial _x^2u)\) the equation \(Pu=f\) is reduced to
Then S given by (2.3) symmetrizes A, and T given by (2.12) diagonalizes S. So we set \(V=T^{-1}U\) and rewrite the equation (3.2) to
where \(A^T=T^{-1}AT\). To simplify notation let us write (3.3) with \(f=0\) as
with \({\mathcal A}=A^T\) and \({\mathcal B}=(\partial _tT^{-1})T-{\mathcal A}(\partial _xT^{-1})T\).
3.1 Energy with scalar weight
Consider an energy with a scalar weight \(\phi (t,x)>0\) with \(\partial _t\phi =1\) and \(|\partial _x\phi |\preceq 1\);
where \(\langle {V,W}\rangle \) stands for the inner product in \({\mathbb {C}}^3\) and \(N>0\) is a positive parameter. In what follows we assume that V(t, x) has small support in x. Note that
Since \(\Lambda {\mathcal A}\) is symmetric and hence
As for a scalar weight \(\phi \) we assume
Lemma 3.1
The assumption (3.5) implies
Proof
In view of Proposition 2.1 the assertion \(\phi ^2\preceq \lambda _1\) is clear. Note that from \(|\partial _tq(\lambda _i)|\preceq (|\partial _ta|+|b||\partial _tb|)\lambda _i+|\partial _t\Delta |\) and Lemma 2.1 it follows that
Taking (2.14) into account one has
which implies \(\phi |\partial _t\lambda _1|\preceq \lambda _1\) thanks to (3.5) and (2.4). As for \(\lambda _2\) noting that \(|\partial _t\Delta |\preceq a^2\) by Lemma 2.1 the assertion follows immediately from (3.5) and (2.5). \(\square \)
3.2 Estimate of energy, terms \(\langle {(\partial _t\Lambda )V,V}\rangle \), \(\langle {(\partial _x\phi )\Lambda {\mathcal A}V,V}\rangle \)
Thanks to (3.6), \(\big |\phi ^{-N}\langle {(\partial _t\Lambda )V,V}\rangle \big |\) is bounded by \(N\phi ^{-N-1}\langle {\Lambda V,V}\rangle \) taking N large. On the other hand, from Lemma 3.2 below it follows that
Recalling that \(\Lambda {\mathcal A}\) is symmetric it is clear that \(\big |\langle {\Lambda {\mathcal A}V,V}\rangle \big |\) is bounded by
Since \(\lambda _1\preceq \sqrt{\lambda _1}\,a\) and hence \(\lambda _1|V_1|\,|V_2|\preceq \sqrt{a}\,(\lambda _1|V_1|^2+a\,|V_2|^2)\) it follows that
with some \(C>0\). In a small neighborhood of (0, 0) where a is enough small one can bound the right-hand side by \(N\phi ^{-N-1}\langle {\Lambda V,V}\rangle \).
3.3 Estimate of energy, term \(\langle {\Lambda {\mathcal B}V,V}\rangle \)
Recall
Applying Lemmas 2.2 and 2.3 we estimate \((\partial _tT^{-1})T\) and \((\partial _xT^{-1})T\). First note that
and \(\langle {\partial _t \mathbf{t}_i,\mathbf{t}_j}\rangle =-\langle {\mathbf{t}_i, \partial _t\mathbf{t}_j}\rangle =-\langle { \partial _t \mathbf{t}_j,\mathbf{t}_i}\rangle \) so that \((\partial _tT^{-1})T\) is antisymmetric. Note that
because \(\sum _{k=1}^3\ell _{ki}\,{\bar{\ell }}_{kj}=0\) if \(i\ne j\). Thanks to Proposition 2.1 and Lemmas 2.3, 2.1 it follows that
and that
Therefore taking (2.12), (3.8) and (3.10) into account one obtains
In order to estimate \(\big |\phi ^{-N}\langle {\Lambda (\partial _tT^{-1})TV,V}\rangle \big |\), noting \(\Lambda \simeq \mathrm{diag}(\lambda _1,a,1)\) and \(\lambda _1\preceq a^2\), it suffices to estimate
Note that
because \(\phi /\lambda _1\preceq 1/\phi \) by (3.6). As for \(|V_1||V_3|\) one has
Finally since \(\sqrt{a}|V_2||V_3|\preceq a |V_2|^2+|V_3|^2\) one concludes that \(\big |\phi ^{-N}\langle {\Lambda (\partial _tT^{-1})TV,V}\rangle \big |\) is bounded by \(N\phi ^{-N-1}\langle {\Lambda V,V}\rangle \) taking N large.
Turn to \(\big |\phi ^{-N}\langle {\Lambda {\mathcal A}(\partial _xT^{-1})TV, V}\rangle \big |\). From Proposition 2.1 and Lemmas 2.2, 2.1 one has
from which one concludes
and hence
Here note that
Lemma 3.2
One has
Proof
With \({\mathcal A}=({\tilde{a}}_{ij})\) it is clear that
Since \(t_{2i}t_{1j}=O(\sqrt{a})\) unless \((i,j)=(2,3)\) and \(t_{3i}t_{2j}=O(\sqrt{a})\) unless \((i,j)=(1,2)\) the first assertion follows from (2.12), (3.15) and (2.11). Note that \(\partial _x\big (1/d_j\big )=O(1/a^{3/2})\) for \(j=1,2\) and \(\partial _x\big (1/d_3\big )=O(1)\) then the second assertion follows from (3.13), (3.15) and (2.12). \(\square \)
From Lemma 3.2 and (3.14) one has
Then it is clear that \(\big |\langle {\Lambda {\mathcal A}(\partial _xT^{-1})TV, V}\rangle \big |\) is bounded by
where
hence \(\big |\phi ^{-N}\langle {\Lambda {\mathcal A}(\partial _xT^{-1})TV, V}\rangle \big |\) is bounded by \(N\phi ^{-N-1}\langle {\Lambda V,V}\rangle \) taking N large.
3.4 Estimate of energy, term \(\langle {\partial _x(\Lambda {\mathcal A})V,V}\rangle \)
Write
and estimate each term on the right-hand side. To estimate the first term note
Lemma 3.3
One has
Proof
Recall
where Lemma 2.1 shows that
because \(1/\lambda _1\preceq a/\Delta \) by Proposition 2.1. Noting that \(\Delta \ge 0\) in a neighborhood of \(x=0\) we see that \(|\partial _x\Delta |\preceq \sqrt{\Delta }\), hence the first inequality. The second inequality follows from the first one thanks to the assumption (3.5). \(\square \)
Since \(|\partial _x\lambda _2|/\lambda _2=O(1/\sqrt{a})\) and \(|\partial _x\lambda _3|/\lambda _3=O(1)\) by Lemma 2.2 it follows from 3.3 that \(\phi (\partial _x\Lambda )=\phi \Lambda (\Lambda ^{-1}\partial _x\Lambda )=\mathrm{diag}\big (O(\lambda _1/\sqrt{a}),O(\phi \sqrt{a}),O(\phi )\big )\) for \(\lambda _2\simeq a\) then by Lemma 3.2 and (3.13) one sees
Therefore to estimate \(\phi \langle {(\partial _x\Lambda ){\mathcal A}V,V}\rangle \) it suffices to bound
Since \(\lambda _1\preceq a\sqrt{\lambda _1}\) and \(\phi \preceq \sqrt{\lambda _1}\) this is bounded by \(\langle {\Lambda V,V}\rangle \) and hence
As for \(\big |\phi ^{-N}\langle {(\partial _x{\mathcal A})V,\Lambda V}\rangle \big |\) it follows from Lemma 3.2 that
hence repeating similar arguments it is easy to see that
which is bounded by \(N\phi ^{-N-1}\langle {\Lambda V,V}\rangle \) taking N large.
Remark 3.1
It shoud be remarked that the condition (3.5), assumed in this section, are stated in terms of \(\Delta \) and a, where a is constant times the discriminant of \(\partial p/\partial \tau \), without any reference to characteristic roots \(\tau _i\).
We conclude this section with an important remark. To obtain energy estimates it suffices to find a finite number of pairs \((\phi _j,\omega _j)\), where \(\phi _j\) is a scalar weight satisfying (3.5) in subregion \(\omega _j\) of which union covers a neighborhood of (0, 0), such that one can collect such estimates obtained in \(\omega _j\). In the following sections this observation is carried out for hyperbolic operators with effectively hyperbolic critical points with coefficients depending on t, also for third order hyperbolic operators with effectively hyperbolic critical points with two independent variables.
4 Hyperbolic operators with effectively hyperbolic critical points with time dependent coefficients
In [9], Ivrii and Petkov proved that if the Cauchy problem (1.1) is \(C^{\infty }\) well posed for any lower order term then the Hamilton map \(F_p\) has a pair of non-zero real eigenvalues at every critical point ([9, Theorem 3]). Here the Hamilton map \(F_p\) is defined by
and a critical point \((X,\Xi )\) is the point where \(\partial p/\partial X=\partial p/\partial \Xi =0\). Note that \(p(X,\Xi )=0\) at critical points by the homogeneity in \(\Xi \) so that \((X,\Xi )\) is a multiple characteristic and \(\tau \) is a multiple characteristic root. A critical point where the Hamilton map \(F_p\) has a pair of non-zero real eigenvalues is called effectively hyperbolic ([7]). In [10], Ivrii has proved that if every critical point is effectively hyperbolic, and p admits a decomposition \(p=q_1q_2\) with real smooth symbols \(q_i\) near the critical point then the Cauchy problem is \(C^{\infty }\) well-posed for every lower order term. In this case the critical point is effectively hyperbolic if and only if the Poisson bracket \(\{q_1,q_2\}\) does not vanish. He has conjectured that the assertion would hold without any additional condition.
If a critical point \((X,\Xi )\) is effectively hyperbolic then \(\tau \) is a characteristic root of multiplicity at most 3 ([9, Lemma 8.1]). If every multiple characteristic root is at most double, the conjecture has been proved in [10,11,12,13, 21, 25, 28]. When there exists an effectively hyperbolic critical point \((X,\Xi )\) such that \(\tau \) is a triple characteristic root, the conjecture is not yet proved in general. For more details about the subsequent progress on proving the conjecture, see [2, 3, 26, 29].
If \((t, x,\tau ,\xi )\) with \(t>0\) is effectively hyperbolic critical point then \(\tau \) must be a double characteristic root ([9, Lemma 8.1]). Therefore a main concern is double or triple characteristic roots at \((0, x, \xi )\). If \(\tau \) is a triple characteristic root at \((0, x, \xi )\) then \((0, x,\tau , \xi )\) is always a critical point. On the other hand, for double characteristic root \(\tau \), the point \((0, x, \tau , \xi )\) is not necessarily a critical point. Here is a simple example
where \(c\in {\mathbb {R}}\). Let \(c\ne 0\) then it is clear that \(\tau =0\) is a double characteristic root at (0, 0, 1). If \(\ell =1\) then \(\partial _tp(0,0,0,1)=-c\ne 0\) and hence (0, 0, 0, 1) is not a critical point. If \(\ell \ge 2\) then (0, 0, 0, 1) is a critical point and \(F_p\) has non-zero real eigenvalues there if and only if \(\ell =2\). Let \(c=0\) then \(\tau =0\) is a triple characteristic root at (0, 0, 1) hence (0, 0, 0, 1) is a critical point and \(F_p\) has non-zero real eigenvalues there if and only if \(\ell = 1\).
Now we restrict ourselves to the case that the coefficients depend only on t and consider the Cauchy problem
where \(a_{j,\alpha }(t)\) (\(j+|\alpha |=m\)) are real valued and \(C^{\infty }\) in \((-c,T)\) with some \(c>0\) and the principal symbol p is hyperbolic for \(t\ge 0\), that is
has only real roots in \(\tau \) for any \((t,\xi )\in [0,T)\times {\mathbb {R}}^n\). Denoting by \({\mathcal S}({\mathbb {R}}^n)\) the set of all rapidly decreasing functions on \({\mathbb {R}}^n\) we have
Theorem 4.1
If every critical point \((0,\tau ,\xi )\), \(\xi \ne 0\) is effectively hyperbolic then there exists \(\delta >0\) such that for any \(a_{j,\alpha }(t)\) with \(j+|\alpha |\le m-1\), which are \(C^{\infty }\) in a neighborhood of \([0,\delta ]\), one can find \(N\in {\mathbb {N}}\), \(C>0\) such that for any \(u_j(x)\in {\mathcal S}({\mathbb {R}}^n)\) \((j=0,\ldots ,m-1)\) there exists a unique solution \(u(t,x)\in C^{\infty }([0,\delta ];{\mathcal S}({\mathbb {R}}^n))\) to the Cauchy problem (4.1) with \(T=\delta \) satisfying
for \(0\le t\le \delta \).
Remark 4.1
Under the assumption of Theorem 4.1 it follows that p has necessarily non-real characteristic roots in the \(t<0\) side near \((0,\xi )\), that is P would be a Tricomi type operator. Indeed from [9, Lemma 8.1] it follows that \(F_p(0,\tau ,\xi )=O\) if all characteristic roots are real in a full neighborhood of \((0,\xi )\) and \(\tau \) is a triple characteristic root.
4.1 Triple effectively hyperbolic characteristic
Assume that \(p(t,\tau ,\xi )\) has a triple characteristic root \({\bar{\tau }}\) at \((0,{\bar{\xi }})\), \(|{\bar{\xi }}|=1\) and \((0,{\bar{\tau }},{\bar{\xi }})\) is effectively hyperbolic. As we see later, without restrictions one may assume that \(m=3\) and p has the form
where \(a(t,\xi )\) and \(b(t,\xi )\) are homogeneous of degree 0 in \(\xi \) and satisfy
The triple characteristic root of \(p(0,\tau ,{\bar{\xi }})=0\) is \(\tau =0\) and
hence \((0,0,{\bar{\xi }})\) is effectively hyperbolic if and only if
Since \(a(0,{\bar{\xi }})=0\) and \(\partial _ta(0,{\bar{\xi }})\ne 0\) there is a neighborhood \({\mathcal U}\) of \((0,{\bar{\xi }})\) in which one can write
where \(e_1>0\) in \({\mathcal U}\). Note that \(\alpha (\xi )\ge 0\) near \({\bar{\xi }}\) because \(a(t,\xi )\ge 0\) in \([0,T)\times {\mathbb {R}}^n\).
Lemma 4.1
There exists a neighborhood \({\mathcal U}\) of \((0,{\bar{\xi }})\) in which one can write
where \(e_2>0\) and \(a_j({\bar{\xi }})=0\).
Proof
Thanks to the Malgrange preparation theorem it suffices to show
It is clear that \(\partial _t^ka^3=0\) at \((0,{\bar{\xi }})\) for \(k=0,1,2\) and \(\partial _t^3a(0,{\bar{\xi }})\ne 0\). Since \(\Delta =4a^3-27b^2\) and \(b(0,{\bar{\xi }})=0\) it is enough to show \(\partial _tb(0,{\bar{\xi }})=0\). Suppose \(\partial _tb(0,{\bar{\xi }})\ne 0\) and hence
where \(b_1\ne 0\). Since \(a(t,{\bar{\xi }})=c\,t\) with \(c>0\) then \(\Delta (t,{\bar{\xi }})=4\,c^3\,t^3-27b(t,{\bar{\xi }})^2\ge 0\) is impossible. This proves the assertion. \(\square \)
Lemma 4.2
There exist a neighborhood U of \({\bar{\xi }}\) and a positive constant \(\varepsilon >0\) such that for any \(\xi \in U\) one can find \(j\in \{1,2,3\}\) such that
where \(\nu _j(\xi )\) are roots of \(t^3+a_1(\xi )t^2+a_2(\xi )t+a_3(\xi )=0\).
Proof
In view of Lemma 4.1 one can write
where \({\hat{b}}=3\sqrt{3}\,b/(2e_1^{3/2})\) and hence
with \(E=e_2/(4e_1^3)\). Choose \(I=[-\delta _1,\delta _1]\), \(\delta _1>0\) and a neighborhood \(U_1\) of \({\bar{\xi }}\) such that \(I\times U_1\subset {\mathcal U}\) and denote
Write \({\hat{b}}(t,\xi )={\hat{b}}_0(\xi )+{\hat{b}}_1(\xi )t+{\hat{b}}_2(\xi )t^2+{\hat{b}}_3(t,\xi )t^3\) and set
Choose a neighborhood \(U\subset U_1\) of \({\bar{\xi }}\) such that
which is possible because \(\alpha ({\bar{\xi }})=0\). Choose \(\varepsilon =\varepsilon (B,C)>0\) such that
and prove that if there exists \(\xi \in U\) such that
we would have a contradiction. We omit to write \(\xi \) for simplicity. Recall
and hence, taking \(t=0\) one has \(\alpha ^3-{\hat{b}}_0^2=E(0)|\nu _1\nu _2\nu _3|<C\varepsilon ^3\alpha ^3\). This shows
Differentiating (4.10) by t and putting \(t=0\) one has
This gives
In view of (4.11) one has
Differentiating (4.10) twice by t and putting \(t=0\) on has
which proves \(\big |4{\hat{b}}_0{\hat{b}}_2+2{\hat{b}}_1^2\big |\ge 6\,\alpha \big (1-\varepsilon C-\varepsilon ^2CB-\varepsilon ^3CB^2/6\big )\). Using (4.12) one obtains
where the right-hand side is greater than \(\alpha \) by (4.8). On the other hand from (4.7) and (4.11) we have
and hence \(4\,B\sqrt{\alpha }> 1\) which contradicts with (4.9). \(\square \)
Denote \(\Delta /e_2\) by \({\bar{\Delta }}\);
Lemma 4.3
There is a neighborhood \(V\subset U\) of \({\bar{\xi }}\) where one can write either
or
Proof
Let \(\nu _j(\xi )\), \(j=1,2,3\) be the roots of \({\bar{\Delta }}(t,\xi )=0\). Since \(\nu _j({\bar{\xi }})=0\) one can assume \(|\nu _j(\xi )|<\delta _1\) in V. Since \(a_j(\xi )\) are real we have two cases; one is real and other two are complex conjugate or all three are real. For the former case denoting the real root by \(\nu _1(\xi )\) we have (4.13) where \(\nu _1(\xi )\le 0\) because \({\bar{\Delta }}\ge 0\) for \(0\le t\le \delta _1\). In the latter case, if two of them coincide, denoting the remaining one by \(\nu _1(\xi )\) one has (4.13). If \(\nu _j(\xi )\) are different each other then we have (4.14) since \({\bar{\Delta }}\ge 0\) for \(0\le t\le \delta _1\). \(\square \)
4.2 Key proposition
Thanks to Lemma 4.3 we have either (4.13) or (4.14). As was observed in [22] (see also [27]) in order to obtain energy estimates it is important to consider not only real zeros of \(\Delta \) but also \({\mathsf {Re}}\,\nu _j(\xi )\), the real part of \(\nu _j(\xi )\) for non-real zeros. Define
with small \(\delta >0\). If \(\psi (\xi )\le 0\), we have \(\phi _1=\phi _2=t\) and \(\omega _2=[0,\delta ]\). The next proposition is the key to applying the arguments in Sect. 3 to operators with triple effectively hyperbolic characteristics.
Proposition 4.1
There exist a neighborhood U of \({\bar{\xi }}\), positive constants \(\delta >0\) and \(C>0\) such that
for any \(\xi \in U\) and \(t\in \omega _j(\xi )\), \(j=1,2\).
Proof
Thanks to (4.6) and Lemma 4.1 it suffices to prove (4.15) for \({\bar{\Delta }}\) and \(t+\alpha \) instead of \(\Delta \) and a. Note that
in the case (4.13) and (4.14) respectively. Since \(|\phi _2|\ge t=\phi _1\) in \(\omega _1(\xi )\) and \(t=\phi _1\ge |\phi _2|\) in \(\omega _2(\xi )\) it is easy to see
for both cases. Similarly noting \(t+\alpha (\xi )\ge t\) it is clear that
Therefore it rests to prove \(\phi _j^2\,(t+\alpha )\le C{\bar{\Delta }}\) in \(\omega _j(\xi )\). First we study the case (4.13). From Lemma 4.2 either \(\varepsilon ^{-1}|\nu _1|\ge \alpha \) or \(\varepsilon ^{-1}|\nu _2|\ge \alpha \) holds. First assume that \(\varepsilon ^{-1}|\nu _1|\ge \alpha \) and hence \(t+|\nu _1|\ge \varepsilon (t+\alpha )\) then
Next assume \(\varepsilon ^{-1}|\nu _2|\ge \alpha \). If \(0<{\mathsf {Re}}\,\nu _2\le |{\mathsf {Im}}\,\nu _2|\) one has for \(t\ge 0\)
which also holds for \({\mathsf {Re}}\,\nu _2\le 0\) clearly. Then we see that
Therefore it follows that
If \({\mathsf {Re}}\,\nu _2>|{\mathsf {Im}}\,\nu _2|\) noting that, for \( t\in \omega _1\)
one has \((4+\varepsilon )|t-{\mathsf {Re}}\,\nu _2|\ge \varepsilon (t+\alpha )\) in \(\omega _1\). Hence
For \(t\in \omega _2\) note that
and hence \((4+\varepsilon )\,t\ge \varepsilon (t+\alpha )\). Thus one has
which proves the assertion for the case (4.13). In the case (4.14) note that
If \(\varepsilon ^{-1}|\nu _j|\ge \alpha \) then \(t+|\nu _j|\ge \varepsilon (t+\alpha )\) and hence it is clear that
which shows the assertion. Thus the proof of Proposition 4.1 is completed. \(\square \)
4.3 Energy estimates
Let P be a differential operator of order 3 with coefficients depending on t. After Fourier transform in x the equation \(Pu=f\) reduces to
where \({\hat{u}}(t,\xi )\) stands for the Fourier transform of u(t, x) with respect to x. With
it is clear that \({\hat{v}}=E(t,\xi ){\hat{u}}\) satisfies
where \(b_j(t,\xi )=0\) for \(|\xi |\le 1\) can be assumed since energy estimates for \(|\xi |\le 1\) is easily obtained. Since
in order to obtain energy estimates for \({\hat{u}}\) one can assume that \({\hat{u}}\) satisfies (4.17) from the beginning. With \(U={^t}\big (D_t^2{\hat{u}},|\xi |D_t{\hat{u}}, |\xi |^2{\hat{u}}\big )\) the equation (4.17) can be written
Let \(S(t,\xi )\) and \(T(t,\xi )\) be defined in Sect. 2 with \(X=\xi \) such that \(T^{-1}ST=\Lambda =\mathrm{diag}\,(\lambda _1,\lambda _2,\lambda _3)\). With \(V=T^{-1}U\) one has
where \({\mathcal A}=T^{-1}AT\) and \({\mathcal B}=T^{-1}BT+ (\partial _tT^{-1})T\). Thanks to Proposition 4.1 we have candidates for scalar weights in each \(\omega _j\). To simplify notation denote
and following [22] (also [27]) introduce three subintervals \(\Omega _j=[t_{j-1}(\xi ),t_j(\xi )]\) and scalar weights \(\varphi _j\), \(j=1,2,3\)
Note that \(\omega _1=\Omega _1\), \(\omega _2=\Omega _2\cup \Omega _3\) and \(\varphi _j=|\phi _2|\) in \(\Omega _j\), \(j=2,3\). Thanks to Proposition 4.1 and Lemma 3.1 one has
where C is independent of \(\xi \in U\). Consider the following energy in \(\Omega _j(\xi )\);
Since \({\mathsf {Re}}\,\langle {i\Lambda {\mathcal A}|\xi |V,V}\rangle =0\) and \(\partial _t\varphi _j=(-1)^{j-1}\) one has
Repeating the same arguments as in Sect. 3 one can estimate
in \(\Omega _j\). It rests to estimate \(|g_j\langle {\Lambda B^TV,V}\rangle |\). Let \(C'\) be a bound of all entries of \(B^T\). Since \(0\le \lambda _1\le \lambda _2\le \lambda _3\) then \(|g_j\langle {\Lambda B^TV,V}\rangle |\) is bounded by
Thanks to (4.19) one has
and
and therefore there exists \(C_2\) such that
Noting that
and \(|\langle {\Lambda {\tilde{F}},{\tilde{F}}}\rangle |\le C''\Vert {\tilde{F}}\Vert ^2=C''\Vert F\Vert ^2\) we obtain
Lemma 4.4
Let \(j=1\) or 3. There exist \(N_0\) and \(C>0\) such that for any \(N\ge N_0\) and any \(U(t,\xi )\) verifying \(\partial _t^kU(t_{j-1}(\xi ),\xi )=0\), \(k=0,1,\ldots , N\) one has
for \(t\in \Omega _j(\xi )\).
With \(\langle {\xi }\rangle =|\xi |+1\) it follows from Lemma 4.4 that
Corollary 4.1
Let \(j=1\) or 3. There is \(N_0\) such that for any \(L\in {\mathbb {N}}\) there exists \(C_L>0\) such that for any U with \(\partial _t^kU(t_{j-1}(\xi ),\xi )=0\), \(k=0,1,\ldots ,N+L\) one has
for \(t\in \Omega _j(\xi )\) and \(N\ge N_0\).
For the subinterval \(\Omega _2(\xi )\) the argument in Sect. 3 shows again
Lemma 4.5
There exist \(N_0\in {\mathbb {N}}\) and \(C>0\) such that one has
for \(t\in \Omega _2(\xi )\) and \(N\ge N_0\).
Corollary 4.2
There exists \(N_0\) such that for any \(L\in {\mathbb {N}}\) there is \(C_L\) such that
for \(t\in \Omega _2\) and \(N\ge N_0\).
Since energy estimates in each subinterval \(\Omega _j\) is obtained, repeating the same arguments as in [22, 27, Sect. 6] one can collect the energy estimates in \(\Omega _j\) yielding energy estimates of \(U(t,\xi )\) in the whole interval \([0,\delta ]\).
Proposition 4.2
Assume that p has a triple characteristic root \({\bar{\tau }}\) at \((0,{\bar{\xi }})\), \(|\bar{\xi }|=1\) and \((0,{\bar{\tau }},{\bar{\xi }})\) is effectively hyperbolic. Then there exist \(\delta >0\) and a conic neighborhood U of \({\bar{\xi }}\) such that for any \(a_{j,\alpha }(t)\) with \(j+|\alpha |\le 2\) one can find \(N_0\in {\mathbb {N}}\) such that for any \(q\in {\mathbb {N}}\) with \(q\ge N_0\) there is \(C>0\) such that
for \((t,\xi )\in [0,\delta ]\times U\) and for any \({\hat{u}}(t,\xi )\) with \(\partial _t^k{\hat{u}}(0,\xi )=0\), \(k=0,1,2\) and \({\hat{f}}(t,\xi )\) with \(\partial _t^k{\hat{f}}(0,\xi )=0\), \(k=0,\ldots ,q+N_0\) satisfying (4.16).
4.4 Remarks on double characteristics
Assume that P is a differential operator of order 2 and the principal symbol p has a double characteristic root \({\bar{\tau }}\) at \((0,{\bar{\xi }})\), \(|\xi |=1\). After Fourier transform in x the equation \(Pu=f\) reduces to
Making similar procedure in Sect. 4.3 one can assume that the principal symbol p has the form
so that \({\bar{\tau }}=0\) is a double characteristic root.
If \(\partial _ta(0,{\bar{\xi }})\ne 0\) one can write
in some neighborhood \({\mathcal U}\) of \((0,{\bar{\xi }})\) where \(e_1>0\) and \(\alpha (\xi )\ge 0\) near \({\bar{\xi }}\). In this case we choose \( \varphi _1=t\), \(\Omega _1=[0,\delta ]\) so that \(\varphi _2\) is not needed.
If \((0,0,{\bar{\xi }})\) is a critical point (hence \(\partial _ta(0,{\bar{\xi }})= 0\)) and effectively hyperbolic then
Indeed, assuming \(a(0,{\bar{\xi }})=\partial _ta(0,{\bar{\xi }})=0\) it is easy to see
which shows that \(\partial _t^2a(0,{\bar{\xi }})\ne 0\) if \((0,0,{\bar{\xi }})\) is effectively hyperbolic. From the Malgrange preparation theorem one can write, in some neighborhood \({\mathcal U}\) of \((0,{\bar{\xi }})\)
where \(e_2>0\) and \(a_i({\bar{\xi }})=0\). Note that if \({\mathsf {Re}}\,{\nu }_1(\xi )\ne {\mathsf {Re}}\,{\nu }_2(\xi )\) then \({\nu }_i(\xi )\) is necessarily real and \({\nu }_i(\xi )\le 0\). In the case that either \({\mathsf {Re}}\,{\nu }_1(\xi )\ne {\mathsf {Re}}\,{\nu }_2(\xi )\) or \({\mathsf {Re}}\,{\nu }_1(\xi )={\mathsf {Re}}\,{\nu }_2(\xi )\le 0\) we take \(\varphi _1=t\), \(\Omega _1=[0,\delta ]\) and \(\varphi _2\) is absent. In the case \({\mathsf {Re}}\,{\nu }_1(\xi )={\mathsf {Re}}\,{\nu }_2(\xi )=\psi (\xi )> 0\) so that
we take \(\varphi _1=\psi (\xi )-t\), \(\Omega _1(\xi )=[0,\psi (\xi )]\), \(\varphi _2=t-\psi (\xi )\), \(\Omega _2(\xi )=[\psi (\xi ),\delta ]\). Repeating similar arguments as in [22, 23] one obtains
Proposition 4.3
Assume that p has a double characteristic root \({\bar{\tau }}\) at \((0,{\bar{\xi }})\), \(|\bar{\xi }|=1\) and \((0,{\bar{\tau }}, {\bar{\xi }})\) is effectively hyperbolic if it is a critical point. Then one can find \(\delta >0\) and a conic neighborhood U of \({\bar{\xi }}\) such that for any \(a_{j,\alpha }(t)\) with \(j+|\alpha |\le 1\) one can find \(N_0\in {\mathbb {N}}\) such that for any \(q\in {\mathbb {N}}\) with \(q\ge N_0\) there is \(C>0\) such that
for \((t,\xi )\in [0,\delta ]\times U\) and for any \({\hat{u}}(t,\xi )\) with \(\partial _t^k{\hat{u}}(0,\xi )=0\), \(k=0,1\) and \({\hat{f}}(t,\xi )\) with \(\partial _t^k{\hat{f}}(0,\xi )=0\), \(k=0,\ldots ,q+N_0\) satisfying (4.25).
4.5 Proof of Theorem 4.1
We turn to the Cauchy problem (4.1). First note that, after Fourier transform in x, the equation is reduced to
Proposition 4.4
Assume that every critical point \((0,\tau ,\xi )\), \(\xi \ne 0\) is effectively hyperbolic. Then there exists \(\delta >0\) such that for any \(a_{j,\alpha }(t)\) with \(j+|\alpha |\le m-1\) one can find \(N_0, N_1\in {\mathbb {N}}\) and \(C>0\) such that
for \((t,\xi )\in [0,\delta ]\times {\mathbb {R}}^n\) and for any \({\hat{u}}(t,\xi )\) with \(\partial _t^k{\hat{u}}(0,\xi )=0\), \(k=0,\ldots ,m-1\) and \({\hat{f}}(t,\xi )\) with \(\partial _t^k{\hat{f}}(0,\xi )=0\), \(k=0,\ldots ,N_1\) satisfying \(P(t,D_t,\xi ){\hat{u}}={\hat{f}}\).
Proof
Let \({\bar{\xi }}\ne 0\) be arbitrarily fixed. Write \( p(0,\tau ,{\bar{\xi }})=\prod _{j=1}^r\big (\tau -\tau _j)^{m_j}\) where \(\sum m_j=m\) and \(\tau _j\) are real and different each other, where \(m_j\le 3\) which follows from the assumption. There exist \(\delta >0\) and a conic neighborhood U of \({\bar{\xi }}\) such that one can write
for \((t,\xi )\in (-\delta ,\delta )\times U\) where \(a_{j,k}(t,\xi )\) are real valued, homogeneous of degree k in \(\xi \) and \(p^{(j)}(0,\tau ,{\bar{\xi }})=(\tau -\tau _j)^{m_j}\). If \((0,\tau _j,{\bar{\xi }})\) is a critical point of p, and necessarily \(m_j\ge 2\), then \((0,\tau _j,{\bar{\xi }})\) is a critical point of \(p^{(j)}\) and it is easy to see
with some \(c_j\ne 0\) and hence \(F_{p^{(j)}}(0,\tau _j,{\bar{\xi }})\) has non-zero real eigenvalues if \(F_{p}(0,\tau _j,{\bar{\xi }})\) does and vice versa. It is well known that one can write, in some conic neighborhood U of \({\bar{\xi }}\) that
where \(P^{(j)}\) are differential operators in t of order \(m_j\) with coefficients which are poly-homogeneous symbol in \(\xi \) and R is a differential operators in t of order at most \(m-1\) with \(S^{-\infty }\) (in \(\xi \)) coefficients. Note that the principal symbol of \(P^{(j)}\) is \(p^{(j)}\) and hence the assumptions in Propositions 4.2 and 4.3 are satisfied. Therefore thanks to Propositions 4.2 and 4.3 we have
in some conic neighborhood of \({\bar{\xi }}\) and for \(j=1,\ldots ,r\). Then by induction on \(j=1,\ldots , r\) one obtains
where \(h(t)={\hat{f}}(t)-R\,{\hat{u}}(t)\). Note that for any \(k, l\in {\mathbb {N}}\) there is \(C_{k,l}\) such that
Therefore one concludes that
Then the assertion follows from the Gronwall’s lemma. Finally applying a compactness arguments one can complete the proof. \(\square \)
Proof of Theorem 4.1:
Let \(u_j(x)\in {\mathcal S}({\mathbb {R}}^n)\) and hence \({\hat{u}}_j(\xi )\in {\mathcal S}({\mathbb {R}}^n)\). From \(P{\hat{u}}=0\) one can determine \(\partial _t^k{\hat{u}}(0,\xi )\) successively from \({\hat{u}}_j(\xi )\). Take \(N\ge N_1+m\) and define
which is in \(C^{\infty }({\mathbb {R}};{\mathcal S}({\mathbb {R}}^n))\). With \({\hat{f}}=-P{\hat{u}}_N\) it is clear that \(\partial _t^k{\hat{f}}(0,\xi )=0\) for \(k=0,\ldots ,N_1\). Apply Proposition 4.4 to the following Cauchy problem
to obtain
Since it is clear that
for \(0\le s\le \delta \) then noting that \({\hat{u}}={\hat{w}}+{\hat{u}}_N\) is a solution to the Cauchy problem (4.29) one obtains
Therefore, by a Paley-Wiener Theorem we prove the \(C^{\infty }\) well-posedness of the Cauchy problem (4.1). \(\square \)
5 Third order hyperbolic operators with effectively hyperbolic critical points with two independent variables
In this section we consider the Cauchy problem for third order operators with two independent variables in a neighborhood of the origin for \(t\ge 0\);
where the coefficients \(a_{j,k}(t,x)\) (\(j+k=3\)) are assumed to be real valued real analytic in (t, x) in a neighborhood of the origin and the principal symbol p
has only real roots in \(\tau \) for \((t,x)\in [0,T')\times U\) with some \(T'>0\) and a neighborhood U of the origin.
Theorem 5.1
If every critical point \((0,0,\tau ,1)\) is effectively hyperbolic then there exist \(T>0\), \(\delta >0\) such that for any \(a_{j,k}(t,x)\) with \(j+k\le 2\), which are \(C^{\infty }\) in a neighborhood of \(\Omega =\{(t,x)\mid |x|\le \delta (T-t), 0\le t\le T\}\), and for any \(n\in {\mathbb {N}}\) one can find \(Q\in {\mathbb {N}}\), \(C>0\) such that for any \(u_j(x)\in C^{\infty }({\mathbb {R}})\) \((j=0,1)\) there exists a unique solution \(u(t,x)\in C^{\infty }(\Omega )\) to the Cauchy problem (5.1) satisfying
5.1 Key proposition, x dependent case
Assume that p has a triple characteristic root \({\bar{\tau }}\) at (0, 0, 1) hence \((0,0,{\bar{\tau }},1)\) is a critical point. Making a suitable change of local coordinates \(t=t'\), \(x=x(t',x')\) such that \(x(0,x')=x'\) one can assume that \(a_{2,1}(t,x)=0\) so that
Since the triple characteristic root is now \({\bar{\tau }}=0\) hence \(b(0,0)=a(0,0)=0\) and the hyperbolicity condition implies that
Note that \(\partial _xa(0,0)=\partial _tb(0,0)=\partial _xb(0,0)=0\) which follows from Lemma 2.1 then it is clear that
This implies \( \partial _ta(0,0)\ne 0\) since (0, 0, 0, 1) is assumed to be effectively hyperbolic.
A key proposition corresponding to Proposition 4.1 is obtained by applying similar arguments as in Sect. 4.1 together with some observations on non-negative real analytic functions with two independent variables given in [22, Lemma 2.1] (see also [27]). We just give a sketch of the arguments. From the Weierstrass’ preparation theorem there is a neighborhood of (0, 0) where one can write
where \(e_2>0\) and \(a_j(x)\) are real valued, real analytic with \(a_j(0)=0\). Denote
then the next lemma corresponds to Lemma 4.3
Lemma 5.1
There exists \(\delta >0\) such that, in each interval \(0<\pm \,x<\delta \), one can write
where \(\nu _1(x)\) is real valued with \(\nu _1(x)\le 0\) or
where \(\nu _k(x)\) are real valued with \(\nu _k(x)\le 0\), in both cases \(\nu _j(x)\) are expressed as convergent Puiseux series;
on \(0<\pm x<\delta \). In all cases there is \(C>0\) such that
Next we show a counterpart of Lemma 4.2. Note that one can write
where \(e_1>0\) and \(\alpha (x)\) is real analytic with \(\alpha (0)=0\) and \(\alpha (x)\ge 0\) in \(|x|<\delta \).
Lemma 5.2
There exist \(\varepsilon >0\) and \(\delta >0\) such that one can find \(j^{\pm }\in \{1,2,3\}\) such that
Recall that, choosing a smaller \(\delta >0\) if necessary, one can assume that any two of \({\mathsf {Re}}\,\nu _j(x)\), \({\mathsf {Im}}\,\nu _j(x)\), \(\nu (x)\equiv 0\) are either different or coincide in each interval \(0<\pm x<\delta \). Denote
and define
with a small \(T>0\). If \(\psi =0\) then \(\phi _1=\phi _2=t\) and \(\Omega _2=\{(t,x)\mid |x|<\delta , \; 0\le t\le T\}\). Now we have a key proposition corresponding to Proposition 4.1;
Proposition 5.1
There exist \(\delta >0\), \(T>0\) and \(C>0\) such that
for \((t,x)\in \Omega _j\) and \(j=1,2\).
5.2 Energy estimates, x dependent case
With \(U={^t}(D_t^2u,D_xD_tu, D_x^2u)\) the equation (5.1) can be written
Let T(t, x) be the orthonormal matrix introduced in Sect. 2.2 such that with \(V=T^{-1}U\) the equation (5.8) becomes
where
Let \(\omega \) be an open domain in \({\mathbb {R}}^2\) and let \(g\in C^1(\omega )\) be a positive scalar function. Denote by \(\partial \omega \) the boundary of \(\omega \) equipped with the usual orientation. Let
then one has
Here make a remark on the boundary term. Denote
where \(\tau _j(t,x)\), \(j=1,2,3\) are characteristic roots of \(p(t,x,\tau ,1)\).
Lemma 5.3
Let \(\Gamma : [a,b]\ni x\mapsto (f(x),x)\) be a space-like curve, that is
Then one has
Proof
Since \(g(t,x)>0\) is scalar function it suffices to prove
To simplify notation we denote \(\Lambda (f(x),x)\) and \({\mathcal A}(f(x),x)\) by just \(\Lambda \) and \({\mathcal A}\). Noting that \(\Lambda {\mathcal A}={^t}\!{\mathcal A}\Lambda \) one has
Therefore to prove the assertion it suffices to show
that is, the maximal eigenvalue of \({|f'|^2}({^t\!}{\mathcal A}\Lambda {\mathcal A})\) with respect to \(\Lambda \) is at most 1. From \(^t\!{\mathcal A}\Lambda =\Lambda {\mathcal A}\) it follows that
Since \(\tau _j\) are the eigenvalues of \({\mathcal A}\) we see that the eigenvalues of \(|f'|^2{\mathcal A}^2\) is at most \(|f'(x)|^2\,\tau _{max}^2<1\) hence the assertion. \(\square \)
Define \(\Omega =\{(t,x)\mid |x|\le \delta (T-t), 0\le t\le T\}\) and
where \(\delta>0, T>0\) are small such that the lines \(|x|=\delta (T-t)\), \(0\le t\le T\) are space-like. Thanks to Proposition 5.1 and Lemma 5.1 it follows that
Apply (5.10) with
and \(\omega =\omega _j\) then from the arguments in Sect. 3 one obtains
Lemma 5.4
There exist \(N_0\), \(C>0\) such that
for \(N\ge N_0\).
Repeating the same arguments as in [22, Lemma 3.1] (also [27]) one has
Proposition 5.2
There exists \(C>0\) such that for every \(n\in {\mathbb {N}}\) one can find \(N_1\) such that
for any \(N\ge N_1\).
Following the same arguments as in [22, 27] one can collect energy estimates in each \(\omega _j\) to obtain
Proposition 5.3
Assume that p has a triple characteristic root \({\bar{\tau }}\) at (0, 0, 1) and \((0,0,{\bar{\tau }},1)\) is effectively hyperbolic. Then there exist \(T>0, \delta >0\) such that for any \(a_{j,k}(t,x)\in C^{\infty }(\Omega )\), \(j+k\le 2\) one can find \(C>0\) and \(Q\in {\mathbb {N}}\) such that
for any \(U(t,x)\in C^{\infty }(\Omega )\) with \(\partial _t^kU(0,x)=0\), \(k=0,\ldots , Q\).
5.3 Proof of Theorem 5.1
To complete the proof of Theorem 5.1, study the remaining case that p has a double characteristic root at (0, 0, 1).
Proposition 5.4
Assume that p has a double characteristic root \({\bar{\tau }}\) at (0, 0, 1) such that \((0,0,{\bar{\tau }},1)\) is effectively hyperbolic if it is a critical point. Then the same assertion as in Proposition 5.3 holds.
We give a sketch of the proof. Assume that p has a double characteristic root \({\bar{\tau }}\) at (0, 0, 1) and hence, after a suitable change of local coordinates, one can write
where \(p_1=\tau -b(t,x)\xi \), \(p_2=\tau ^2-a(t,x)\xi ^2\) and \(a(0,0)=0\), \(b(0,0)\ne 0\).
If (0, 0, 0, 1) is a critical point of p and hence \(\partial _ta(0,0)=0\) then \(F_p=c\,F_{p_2}\) at (0, 0, 0, 1) with some \(c\ne 0\) and \(\mathrm{det}\big (\lambda I-F_{p_2}(0,0,0,1)\big )=\lambda ^2\big (\lambda ^2-2\,\partial _t^2a(0,0)\big )\) which shows \( \partial _t^2a(0,0)\ne 0\). From the Weierstrass’ preparation theorem one can write
where \(e_2>0\), \({\tilde{a}}_j(0)=0\) and \(\Delta _2\) takes the form, in each \(0<\pm x<\delta \), either (5.4) or (5.5) where \(\nu _1\) is absent. If \(\Delta _2\) has the form (5.5) or (5.4) with \({\mathsf {Re}}\,\nu _2(x)\le 0\) it suffices to take \(\varphi _1=t\), \(\omega _1=\Omega \) (\(\varphi _2\) is not needed). If \(\Delta _2\) takes the form (5.4) with \({\mathsf {Re}}\,\nu _2(x)=\psi (x)> 0\) we choose \(\varphi _1=\psi (x)-t\), \(\omega _1=\{(t,x)\mid |x|\le \delta (T-t), 0\le t\le \psi (x)\}\) and \(\varphi _2=t-\psi (x)\), \(\omega _2=\{(t,x)\mid |x|\le \delta (T-t), \psi (x)\le t\le T\}\).
If \(\partial _ta(0,0)\ne 0\) one can write
where \(e_1>0\) and \(\alpha (x)\ge 0\) in a neighborhood of \(x=0\). In this case we take \(\varphi _1=t\), \(\omega _1=\Omega \) (\(\varphi _2\) is not needed).
Denote
Then repeating the same arguments as in [22, 27] one has
for \(N\ge N_0\) with \(G^{(2)}_j(u)=g_j\big (|\partial _tu|^2+a|\partial _xu|^2\big )dx+ag_j\big (\partial _xu\cdot \overline{\partial _tu}+\overline{\partial _xu}\cdot \partial _tu\big )dt\). On the other hand, since \(P_1\) is a first order differential operator with a real valued b(t, x) it is easy to see that
for \(N\ge N_0\) with \(G^{(1)}_j(u)=g_j|u|^2dx+bg_j|u|^2dt\). Inserting \(u=P_1u\) in (5.15) and \(u=P_2u\) in (5.16) respectively and adding them one obtains
In view of \(b(0,0)\ne 0\) and \(a(0,0)=0\) it is easy to see that \(D_t^2\), \(D_xD_t\) and \(D_x^2\) are linear combinations of \(D_tP_1\), \(D_xP_1\) and \(P_2\) with smooth coefficients modulo first order operators. Since one can write
one concludes from (5.17) that
The rest of the proof is parallel to that of Proposition 5.3.
Proof of Theorem 5.1:
Note that (5.13) implies
Then applying an approximation argument with the Cauchy-Kowalevsky theorem one can conclude the proof of Theorem 5.1.
We restrict ourselves to third order operators in Theorem 5.1 because it seems to be hard to apply the same arguments as in Sect. 4.5 to this case. \(\square \)
6 Example of third order homogeneous equation with general triple characteristics
To show that the same arguments in the previous sections can be applicable to hyperbolic operators with more general triple characteristics, we study the Cauchy problem
in a full neighborhood of (0, 0) in \({\mathbb {R}}^2\). The Cauchy problem (6.1) is uniformly \(C^{\infty }\) well posed near the origin if one can find a neighborhood U of (0, 0) and a small \(\epsilon >0\) such that for any \(|s|<\epsilon \) and any \(u_j(x)\in C^{\infty }(\Omega \cap \{t=s\})\) there is a unique solution to (6.1). Assume that there is a neighborhood \(U'\) of (0, 0) such that \(p(t,x,\tau ,1)=0\) has only real roots in \(\tau \) for \((t,x)\in U'\), that is
which is necessary for the Cauchy problem (6.1) to be well posed near the origin ([17, 20]).
Theorem 6.1
Assume (6.2) and that there exists \(C>0\) such that
holds in a neighborhood of (0, 0). Then the Cauchy problem (6.1) is uniformly \(C^{\infty }\) well posed near the origin.
Note that if both a(t, x) and b(t, x) are independent of t, Theorem 6.1 is a very special case of [30, Theorem 1.1]. If a(t, x) and b(t, x) are independent of x this is also a special case of [4, Theorem 2] and [19, Theorem 1]. Inparticular, in [19], it is proved that, under the first condition of (6.3) (the second condition is a consequence of the first, in this case), the Bézout matrix itself behaves as if it were diagonal.
We give a rough sketch of the proof. If \(\Delta (0,0)>0\) then p is strictly hyperbolic near the origin and the assertion is clear so \(\Delta (0,0)=0\) is assumed from now on and hence \(a(0,0)=0\) by assumption (6.3). Then p has a triple characteristic root at (0, 0, 1). Thanks to Proposition 2.1 the eigenvalues \(\lambda _j\) of the Bézoutian matrix satisfy
for \(\Delta /a\ge a^2/C\). Let \(\phi \) be a scalar function satisfying \(\phi >0\) and \(\partial _t\phi >0\) in \(\omega \) and consider the following energy;
where \(N>0, \gamma >0\) are positive parameters. Note that
and
Since the hyperbolicity condition \(4a^3\ge 27b^2\) is assumed in a full neighborhood of (0, 0) then Lemma 2.1 now states
Suppose that a scalar weight \(\phi \) satisfying the following property is obtained;
Since \(|\partial _tq(\lambda _j)|\preceq (|\partial _ta|+|b||\partial _tb|)\lambda _j+|\partial _t\Delta |\) and \(|\partial _t\Delta |\preceq a^2\,|\partial _ta|\) by (6.3) then
and hence \( |\partial _t\lambda _j|\preceq (|\partial _ta|/a)\lambda _j+a^{3/2}\,\lambda _j+a\,|\partial _ta|\) for \(j=1,2\) which shows that
Thus \(\phi \,|\partial _t\lambda _j|\preceq \partial _t\phi \,\lambda _j\) for \(j=1,2,3\) since \(\lambda _2\simeq a\). The case \(j=3\) is clear from (6.7). This proves that \(\big |\phi ^{-N}\langle {(\partial _t\Lambda )V,V}\rangle \big |\) is bounded by (6.5) taking N large. Next assume that a scalar weight \(\phi \) satisfies
where \(\omega _{\epsilon }\) is a family of regions converging to (0, 0) as \(\epsilon \rightarrow 0\). Recall (3.4);
Thanks to (3.7) the term \(N\big |\phi ^{-N-1}(\partial _x\phi )\langle {\Lambda {\mathcal A}V,V}\rangle \big |\) is bounded by (6.5) in \(\omega _{\epsilon }\) for enough small \(\epsilon \) in virtue of the assumption (6.9). Consider the term \(\langle {\partial _x(\Lambda {\mathcal A})V,V}\rangle =\langle {(\partial _x\Lambda ){\mathcal A}V,V}\rangle +\langle {(\partial _x{\mathcal A})V,\Lambda v}\rangle \). From (3.16) one has \( |\partial _x\lambda _1|\preceq a^{3/2}\). Since \(|\partial _x\lambda _2|=O(\sqrt{a})\) and \(|\partial _x\lambda _3|=O(1)\) it follows from Lemmas 3.2
and then \(|\langle {(\partial _x\Lambda ){\mathcal A}V,V}\rangle |\) is bounded by
As for \(\big |\langle {(\partial _x{\mathcal A})V,\Lambda V}\rangle \big |\), thanks to Lemma 3.2 one has
hence \(|\langle {\Lambda (\partial _x{\mathcal A})V,V}\rangle |\) is bounded by
Then \(|(e^{-\gamma t}\phi ^{-N}\Lambda {\mathcal A}\partial _x V,V)|\) is bounded by (6.5) taking \(\epsilon >0\) small and \(\gamma \) large.
Turn to \({\mathsf {Re}}\,\big (e^{-\gamma t}\phi ^{-N}\Lambda {\mathcal B}V,V\big )\) where \({\mathcal B}=(\partial _tT^{-1})T-{\mathcal A}(\partial _xT^{-1})T\). Using (6.8) it is easy to see
Noting \(|\partial _tb|\preceq \sqrt{a}\,|\partial _ta|\) these estimates improve (3.11) to
Lemma 6.1
Let \((\partial _tT^{-1})T=({\tilde{t}}_{ij})\) then
In order to estimate \(\big |\phi ^{-N}\langle {(\partial _tT^{-1})TV,\Lambda V}\rangle \big |\) recalling \(\Lambda \simeq \mathrm{diag}(a^2,a,1)\) it suffices to estimate
Note that \(a\,{\tilde{t}}_{21}\preceq a^2+\sqrt{a}\,|\partial _ta|\) and
As for \({\tilde{t}}_{31}|V_1||V_3|\preceq (|\partial _ta|+a^{5/2})|V_1||V_3|\) one has
Noting \({\tilde{t}}_{32}|V_2||V_3|\preceq (\sqrt{a}|\partial _ta|+a^2)|V_2||V_3|\preceq a\,|V_2||V_3|\preceq \sqrt{a}(a|V_2|^2+|V_3|^2)\) one concludes that \(\big |\phi ^{-N}\langle {(\partial _tT^{-1})TV,\Lambda V}\rangle \big |\) is bounded by (6.5) taking N large and \(\gamma \ge 1\). Consider \(|\phi ^{-N}\langle {(\partial _xT^{-1})TV,\Lambda {\mathcal A}V}\rangle \). From Lemma 3.2 it follows that
because \(\Lambda {\mathcal A}\) is symmetric and \(\Lambda \simeq \mathrm{diag}(a^2,a,1)\). Taking (3.14) into account this shows that
Thus \(|\langle {(\partial _xT^{-1})TV,\Lambda {\mathcal A}V}\rangle |\) is bounded by
so that \(|{\mathsf {Re}}\,\big (e^{-\gamma t}\phi ^{-N}\Lambda {\mathcal B}V,V\big )|\) is bounded by (6.5) taking N and \(\gamma \) large.
Therefore to obtain energy estimates it suffices to find pairs \((\phi _j,\omega _{j,\epsilon })\) of scalar weight \(\phi _j\) and subregion \(\omega _{j,\epsilon }\) such that (6.7) and (6.9) are verified and \(\cup \, \omega _{j,\epsilon }\) covers a neighborhood of (0, 0) for any fixed small \(\epsilon >0\). Since the choice of \((\phi _j,\omega _{j,\epsilon })\) is exactly same as in [22] (also [27]) we only mention how to choose \(\phi _j\) and \(\omega _{j,\epsilon }\) (\(\phi \) is denoted by \(\rho \) in [22, 27]).
Following [22] one can define a real valued function \(\alpha (t,x)\)
so that \(a(t,x)=\alpha (t,x)^2\) where \(t_i(x)\) has a convergent Puiseux expansion in \(0<\pm x<\delta \) with small \(\delta \) and \({\mathsf {Im}}\,t_i(x)\ne 0\) if \(i\in I_2\). We choose all distinct \(t_k(x)\) in (6.11) and rename them as \(t_1(x),\ldots ,t_m(x)\). Taking \(\delta \) small one can assume that
Define \(\sigma _j(x)\) by
so that \(\sigma _1(x)\le \cdots \le \sigma _m(x)\) in \(|x|<\delta \). Define
with
where the sum is taken over all distinct \(t_i(x)\) in (6.11). Denote by \(\omega _j^{\pm }\) and \(\omega (T)\) the subregions defined by
for small \({\bar{\delta }}>0\), \(T>0\). Here \({\bar{\delta }}>0\) and \(T>0\) play the role of \(\epsilon >0\) in (6.9).
For \(\omega _j\), \(j=1,\ldots ,m\) we take \(\phi =\phi _j^{\pm }(t,x)=\pm (t-\sigma _j(x))\). For \(\omega (T)\) we take \(\phi =\phi _{m+1}=t-s_m(x)\) if \(n\ge 1\). Turn to \(\omega (T)\) with \(n=0\). Without restrictions one can assume \(\alpha >0\) and \(\partial _t\alpha >0\) in \(\omega (T)\) (see [22, Lemma 2.2]) and we take \(\phi =\alpha (t,x)\).
Change history
05 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00208-021-02291-7
References
D’Ancona, P., Spagnolo, S.: Quasi-symmetrization of hyperbolic systems and propagation of the analytic regurality, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 1, 169–185 (1998)
Bernardi, E., Bove, A., Petkov, V.: Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity. J. Hyper. Differ. Equ. 12, 535–579 (2015)
Bernardi, E., Nishitani, T.: Counterexamples to \(C^{\infty }\) well posedness for some hyperbolic operators with triple characteristics, Proc. Japan Acad., 91, Ser. A, 19–24 (2015)
Colombini, F., Orrù, N.: Well-posedness in \(C^{\infty }\) for some weaky hyperbolic equations. J. Math. Kyoto Univ. 39, 399–420 (1999)
Garetto, C., Ruzhansky, M.: Weakly hyperbolic equations with non-analytic coefficients and lower order terms. Math. Ann. 357, 401–440 (2013)
Garetto, C., Ruzhansky, M.: A note on weakly hyperbolic equations with analytic principal part. J. Math. Anal. Appl. 412, 1–14 (2014)
Hörmander, L.: The Cauchy problem for differential equations with double characteristics. J. Anal. Math. 32, 118–196 (1977)
Hörmander, L.: The analysis of linear partial differential operators, vol. III. Springer, Berlin (1985)
Ivrii, V., Petkov, V.: Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well posed, Uspekhi Mat. Nauk, 29, 3–70 (1974), English translation: Russ. Math. Surv., 29, 1–70 (1974)
Ivrii, V.: Sufficient conditions for regular and completely regular hyperbolicity, Tr. Mosk. Mat. Obs., 33, 3–65 (1975), (in Russian), English translation: Trans. Mosc. Math. Soc., 33, 1–65 (1978)
Iwasaki, N.: The Cauchy problem for effectively hyperbolic equations (a special case). J. Math. Kyoto Univ. 23, 503–562 (1983)
Iwasaki, N.: The Cauchy problem for effectively hyperbolic equations (a standard type). Publ. Res. Inst. Math. Sci. 20, 551–592 (1984)
Iwasaki, N.: The Cauchy problem for effectively hyperbolic equations (general case). J. Math. Kyoto Univ. 25, 727–743 (1985)
Jannelli, E.: On the symmetrization of the principal symbol of the hyperbolic equations. Commun. Partial Differ. Equ. 14, 1617–1634 (1989)
Jannelli, E.: The hyperbolic symmetrizer: theory and applications, in: Advances in Phase Space Analysis of Partial Differential Equations, in: Progr. Nonlinear Differential Equations Appl., vol. 78, Birkhäuser Boston, pp. 113–139 (2009)
Jannelli, E., Taglialatela, G.: Homogeneous weakly hyperbolic equations with time dependent analytic coefficients. J. Differ. Equ. 251, 995–1029 (2011)
Lax, P.D.: Asymptotic solutions of oscillatory initial value problem. Duke Math. J. 24, 627–646 (1957)
Lax, P.D., Nirenberg, L.: On stability for difference schemes: a sharp form of Gårding’s inequality. Comm. Pure Appl. Math. 19, 473–492 (1966)
Kinoshita, T., Spagnolo, S.: Hyperbolic equations with non-analytic coefficients. Math. Ann. 336, 551–569 (2006)
Mizohata, S.: The theory of partial differential equations. Cambridge University Press, Cambridge (1973)
Melrose, R.: The Cauchy problem for effectively hyperbolic operators. Hokkaido Math. J. 12, 371–391 (1983)
Nishitani, T.: The Cauchy problem for weakly hyperbolic equations of second order. Commun. Partial Differ. Equ. 5, 1273–1296 (1980)
Nishitani, T.: Sur les opérateurs fortement hyperboliques qui ne dépendent que du temps, Equations aux dérivées partielles hyperboliques et holomorphes, Sémin. Paris Année 1981–82, Exp. No. 4, 23 p. (1982)
Nishitani, T.: On the finite propagation speed of wave front sets for effectively hyperbolic operators. Sci. Rep. College Gen. Ed. Osaka Univ. 32, 1–7 (1983)
Nishitani, T.: Local energy integrals for effectively hyperbolic operators. I, II, J. Math. Kyoto Univ., 24, 623-658 and 659-666 (1984)
Nishitani, T.: The effectively hyperbolic Cauchy problem, in The Hyperbolic Cauchy Problem, Lecture Notes in Math. 1505, Springer-Verlag, pp. 71–167 (1991)
Nishitani, T.: Hyperbolicity of two by two systems with two independent variables. Commun. Partial Differ. Equ. 23, 1061–1110 (1989)
Nishitani, T.: Notes on symmetrization by Bézoutian. Bollettino dell’Unione Matematica Italiana 13, 417–428 (2020)
Nishitani, T., Petkov, V.: Cauchy problem for effectively hyperbolic operators with triple characteristics. J. Math. Pures Appl. 123, 201–228 (2019)
Spagnolo, S., Taglialatela, G.: Homogeneous hyperbolic equations with coefficients depending on one space variable. J. Hyperbolic Differ. Equ. 4, 533–553 (2007)
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number JP20K03679.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Giga.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original online version of this article was revised: to fulfil the author decision to opt for Open Choice.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Nishitani, T. Diagonal symmetrizers for hyperbolic operators with triple characteristics. Math. Ann. 383, 529–569 (2022). https://doi.org/10.1007/s00208-021-02153-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-021-02153-2