Related papers: On the Gevrey strong hyperbolicity

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 $s>6$.

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with a scalar type spectral operator $A$ in a complex Banach space, we find conditions on $A$, formulated exclusively in terms of the location of its spectrum in the…

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…

This paper proposes sG-hyperbolicity as a new tool for studying hyperbolicity on complex manifolds. It demonstrates that this notion leads to a wider class of divisorially hyperbolic manifolds compared to balanced hyperbolicity. We also…

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…

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variable and random stationary ergodic in time. As was proved in [25] and [13] in this case…

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) +…

We consider solutions of the Cauchy problem for semilinear equations with (possibly) different L\'evy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the…

We consider the problem of the strong unique continuation for an elasticity system with general residual stress. Due to the known counterexamples, we assume the coefficients of the elasticity system are in the Gevrey class of appropriate…

We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…

We study the asymptotic behavior of the solutions related to a singularly perturbed q-difference-differential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so…

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\ge 0, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need…

We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t = 0$ with finite…

In mid 60s Bott proved that (1) the index theorem for homogeneous, G-invariant, elliptic differential operators acting in the spaces of sections of induced representations of G over G/H reduces to the Weyl character formula and (2) the…

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We…

In this article, we study the convolution operators $M_k$ with oscillatory kernel, which are related to solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic…

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…

In this paper we analyse the Gevrey well-posedness of the Cauchy problem for weakly hyperbolic equations of general form with time-dependent coefficients. The results involve the order of lower order terms and the number of multiple roots.…

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = \psi(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically…

Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori…