On the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Given the abstract evolution equation $$y'(t)=Ay(t),\ t\in \mathbb{R},$$ 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 not be strongly differentiable, to be strongly infinite differentiable on $\mathbb{R}$. The important case of the equation with a normal operator $A$ in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at $0$, then all of them are strongly infinite differentiable on $\mathbb{R}$.Comment: A correction in Remarks 3.1, a few minor readability improvements. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1708.0506

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

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 complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $\mathbb{R}$. We also reveal certain inherent smoothness improvement effects and show that, if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator $A$ is necessarily bounded. The important particular case of the equation with a normal operator $A$ in a complex Hilbert space follows immediately.Comment: Minor readability improvement. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1803.10038, arXiv:1708.0506

On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance

A new numerical method is introduced to study the problem of time evolution of generic non-linear dynamical systems in four-dimensional spacetimes. It is assumed that the time level surfaces are foliated by a one-parameter family of codimension two compact surfaces with no boundary and which are conformal to a Riemannian manifold C. The method is based on the use of a multipole expansion determined uniquely by the induced metric structure on C. The approach is fully spectral in the angular directions. The dynamics in the complementary 1+1 Lorentzian spacetime is followed by making use of a fourth order finite differencing scheme with adaptive mesh refinement. In checking the reliability of the introduced new method the evolution of a massless scalar field on a fixed Kerr spacetime is investigated. In particular, the angular distribution of the evolving field in to be superradiant scattering is studied. The primary aim was to check the validity of some of the recent arguments claiming that the Penrose process, or its field theoretical correspondence---superradiance---does play crucial role in jet formation in black hole spacetimes while matter accretes onto the central object. Our findings appear to be on contrary to these claims as the angular dependence of a to be superradiant scattering of a massless scalar field does not show any preference of the axis of rotation. In addition, the process of superradiance, in case of a massless scalar field, was also investigated. On contrary to the general expectations no energy extraction from black hole was found even though the incident wave packets was fine tuned to be maximally superradiant. Instead of energy extraction the to be superradiant part of the incident wave packet fails to reach the ergoregion rather it suffers a total reflection which appears to be a new phenomenon.Comment: 49 pages, 11 figure

On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups

Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable C₀-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator.Знайдено необхідні та достатні умови для того, щоб спектральний оператор скалярного типу в банаховому просторі породжував ультрадиференційовну C₀-напівгрупу Карлемана. Ці умови сформульовано виключно у термінах спектра оператора

On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

On nonlinear stabilization of linearly unstable maps

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For G\^ateaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fr\'echet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fr\'echet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.Comment: New section 1.5 and several references added. 20 pages, no figur