Related papers: Notes on tangent bicharacteristics and ill-posedne…

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…

For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In…

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}).…

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…

In this paper we study the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of…

We prove that the Cauchy problem for the model hyperbolic operator in $ \R^{4} $ \[ Q=-D_t^2+2xD_tD_y+D_x^2+x^3D_y^2+D_z^2+z^2D_y^2 \] is not locally solvable at the origin, in the Gevrey $s$ class if $s>6$.

We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have real principal symbol and we shall consider the limits of bicharacteristics at the set where the principal symbol…

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…

We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in…

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…

For first-order quasi-linear systems of partial differential equations, we formulate an assumption of a transition from initial hyperbolicity to ellipticity. This assumption bears on the principal symbol of the first-order operator. Under…

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…

We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap \{0 \leq t \leq T\},\: \Omega \subset \R^{n+1}$ with triple characteristics on $t = 0$. We consider the case when the fundamental matrix of the principal symbol…

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in [24] and [12] in this case…

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of…

In this paper we consider the Cauchy problem for higher order weakly hyperbolic equations. We assume that the principal symbol depends only on one space variable and the characteristic roots $\tau_j$ verify the inequality \[\tau_j^2(x) +…

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 $s$…

We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have complex principal symbol satisfying condition ($\Psi$) and we shall consider the limits of semibicharacteristics at the…

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

The Cauchy problem is investigated for the parabolic type in the some finite part $[t_0, t_1] \subset [0, \infty)$ of the semi axis $t \in [0, \infty)$ and degenarated to Schrodinger type in the remain part of the same semi axes the second…