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For a homogeneous polynomial $p$ in $\xi\in {\bf R}^n$ with Gevrey coefficients, it is known that the Cauchy problem for any realization of $p$ is well-posed in the Gevrey class of order $s<2$ if the characteristic roots are real. In this note, we give examples showing the situation of the converse direction, in particular the optimality of the Gevrey order $s=2$., Comment: 5 pages

Geometrical aspects of effectively hyperbolic singular points over $t=0$ are discussed assuming that the characteristic roots are real only on the one side $t\geq 0$. In particular, the difference from the case that the characteristic roots are real in both sides $t\geq 0$ and $t<0$., Comment: 9 pages

In this note, we improve a previously proven non-solvability result of the Cauchy problem for the Cauchy problem in the Gevrey class for a homogeneous second-order differential operator mentioned in the title. We prove that the Cauchy problem for this operator is not locally solvable at the origin for any lower order term in the Gevrey class of order greater than 5, lowering the previously obtained Gevrey order 6., Comment: 12 pages

This paper is devoted to a simpler derivation of energy estimates compared to previously existing ones, for effectively hyperbolic operators. One of main points is no use of general Fourier integral operators and another point is an efficient use of Weyl calculus of pseudodifferential operators associated with several different metrics., Comment: 31pages

Ivrii's conjecture asserts that the Cauchy problem is $C^{\infty}$ well-posed for any lower order term if every critical point of the principal symbol is effectively hyperbolic. Effectively hyperbolic critical point is at most triple characteristic. If every characteristic is at most double this conjecture has been proved in 1980's. In this paper we prove the conjecture for the remaining cases, that is for operators with triple effectively hyperbolic characteristics., Comment: 63p

Symmetrizers for hyperbolic equations are obtained by diagonalizing the Bezoutian matrix of hyperbolic symbols. Such diagonal symmetrizers are applied to the Cauchy problem for hyperbolic operators with triple characteristics. In particular, the V.Ivrii's conjecture concerned with triple effectively hyperbolic characteristics is proved for differential operators with coefficients depending on the time variable.

Let $p$ be a monic hyperbolic polynomial and let $H$ be the Bezoutian matrix of $p$ and $p'$. Then $H$ symmetrizes the Sylvester matrix associated with $p$. This fact is observed by E.Jannelli. We give a simple proof of this fact and at the same time show that the family of Bezoutian matrices of Nuij approximation of $p$ gives quasi-symmetrizers introduced by S.Spagnolo. A relation connecting $H$with the symmetrizer which was used by J.Leray for strictly hyperbolic polynomial is given.

This paper studies the Cauchy problem for variable coefficient weakly hyperbolic first order systems of partial differential operators. The hyperbolicity assumption is that for each $t, x$ the principal symbol is hyperbolic. No hypothesis is imposed on lower order terms. For coefficients and Cauchy data sufficiently Gevrey regular the Cauchy problem has a unique sufficiently Gevrey regular solution. We prove stability and error estimates for the spectral Crank-Nicholson scheme. Approximate solutions can be computed with accuracy $epsilon$ in the supremum norm with cost growing at most polynomially in $epsilon^{-1}$. The proofs use the symmetrizers from [2].

We study effectively hyperbolic operators $P$ with triple characteristics points lying on $t= 0$. Under some conditions on the principal symbol of $P$ one proves that the Cauchy problem for $P$ in $[0, T] \times U$ is well posed for every choice of lower order terms. Our results improves those in [11] since we don't assume the condition (E) of [11] satisfied.

We consider the Cauchy problem for second order differential operators with two independent variables $P=D_t^2-D_x(b(t)a(x))D_x$. Assume that $b(t)$ is a nonnegative $C^{n,alpha}$ function and $a(x)$ is a nonnegative Gevrey function of order $s>1$ we prove that the Cauchy problem for $P$ is well-posed in the Gevrey class of any order $s

We study the Cauchy problem for effectively hyperbolic operators $P$ with principal symbol $p(t, x,\tau,\xi)$ having triple characteristics on $t = 0$. Under a condition (E) we show that such operators are strongly hyperbolic, that is the Cauchy problem is well posed for $p(t, x,D_t, D_x) + Q(t, x, D_t, D_x)$ with arbitrary lower order term $Q$. The proof is based on energy estimates with weight $t^{-N}$ for a first order pseudo-differential system, where $N$ depends on lower order terms. For our analysis we construct a non-negative definite symmetrizer $S(t)$ and we prove a version of Fefferman-Phong type inequality for ${\rm Re}\, (S(t)U, U)_{L^2({\mathbb R}^n)}$ with a lower bound $-C t^{-1}\|\langle D \rangle^{-1}U\|_{L^2(\mathbb R^n)}$.

We consider the Cauchy problem for first order systems. Assuming that the set of the singular points of the characteristic variety is a smooth manifold and the characteristic values are real and semi-simple we introduce a new class which is strictly hyperbolic in the transverse direction to the characteristic manifold. We prove that if the characteristic manifold is either involutive or symplectic then these transversally strictly hyperbolic systems are strongly hyperbolic. On the other hand if the characteristic manifold is neither involutive nor symplectic transversally strictly hyperbolic systems are much more involved which is discussed taking an interesting example.

We discuss the well-posedness of the Cauchy problem for hyperbolic operators with double characteristics which changes from non-effectively hyperbolic to effectively hyperbolic, on the double characteristic manifold, across a submanifold of codimension 1. We assume that there is no bicharacteristic tangent to the double characteristic manifold and the spatial dimension is 2. Then we prove the well-posedness of the Cauchy problem in all Gevrey classes assuming, on the double characteristic manifold, that the ratio of the imaginary part of the subprincipal symbol to the real eigenvalue of the Hamilton map is bounded and that the sum of the real part of the subprincipal symbol and the modulus of the imaginary eigenvalue of the Hamilton map is strictly positive.

In this paper we are concerned with a homogeneous differential operator $p$ of order $m$ of which characteristic set of order $m$ is assumed to be a smooth manifold. We define the Gevrey strong hyperbolicity index as the largest number $s$ such that the Cauchy problem for $p+Q$ is well-posed in the Gevrey class of order $s$ for any differential operator $Q$ of order less than $m$. We study the case of the largest index and we discuss in which way the Gevrey strong hyperbolicity index relates with behaviors of bicharacteristics of $p$ near the characteristic manifold.

In this paper we prove that for a class of non-effectively hyperbolic operators with smooth triple characteristics the Cauchy problem is well posed in the Gevrey 2 class, beyond the generic Gevrey class $ 3/2 $ (see e.g. \cite{Bro}). Moreover we show that this value is optimal.

In this note we concern with the wave maps from the Lorentzian manifold with the periodic in time metric into the Riemannian manifold, which belongs to the one-parameter family of Riemannian manifolds. That family contains as a special case the Poincare upper half-plane model. Our interest to such maps is motivated with some particular type of the Robertson-Walker spacetime arising in the cosmology. We show that small periodic in time perturbation of the Minkowski metric generates parametric resonance phenomenon. We prove that, the global in time solvability in the neighborhood of constant solutions is not a stable property of the wave maps.