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We study the Cauchy problem for effectively hyperbolic operators $P$ with triple characteristics points lying on the initial plane $t= 0$. Under some conditions on the principal symbol of $P$ one proves that the Cauchy problem for $P$ in $[0, T] \times \Omega \subset {\mathbb R}^{n+1}$ is well posed for every choice of lower order terms. Our results improves those in [11] since we do not assume the condition (E) of [11] to be satisfied.

In this note we study lower bounds of energy growth for solutions to wave equations which are compact in space perturbations of the wave equation $\partial_{t}^{2}u - \Delta u = 0$. Assuming that there exists a null bicharacteristic $(x(t),\xi(t))$, parametrized by the time $t$, such that $x(t)$ remains inside a ball and $\xi(t)$ outside a ball for $t \geq 0$ we prove that the solution operator $R(t)$ is bounded from below by constant times $\sqrt{\lvert\xi(t)\rvert/\lvert\xi(0)\rvert}$ in the operator norm. We apply this result to examples constructed by the same idea as in Colombini and Rauch [1] and show that there exist compact in space perturbations which cause $\exp(ct^{\alpha})$ growth of the energy for any given $0\leq \alpha \leq 1$.

In this paper we prove that for non effectively hyperbolic operators with smooth double characteristics with the Hamilton map exhibiting a Jordan block of size 4 on the double characteristic manifold the Cauchy problem is well posed in the Gevrey 6 class if the strict Ivrii-Petkov-Hörmander condition is satisfied., 2000 Mathematics Subject Classification: 35L15, Secondary 35L30.

In this paper we discuss the $C^{\infty}$ well-posedness for second order hyperbolic equations $Pu=\partial_{t}^{2}u-a(t,x) \partial_{x}^{2}u=f$ with two independent variables $(t,x)$. Assuming that the $C^{\infty}$ function $a(t,x) \geq 0$ verifies $\partial_{t}^{p}a(0,0)\neq 0$ with some $p$ and that the discriminant $\Delta(x)$ of $a(t,x)$ vanishes of finite order at $x=0$, we prove that the Cauchy problem for $P$ is $C^{\infty}$ well-posed in a neighbourhood of the origin.

We consider the Cauchy problem for the operator $D_t^2-D_xa(t,x)D_x$ in the Gevrey classes. We show that if the coefficient $a(t,x)$ is given by a finite sum of non negative functions then the Cauchy problem is well posed in the wider Gevrey class for the larger powers. We also give an example showing that the order of the Gevrey class obtained here is optimal.