Infinity In Space Research Articles

Scattering of a 2D Schrödinger equation with exponential growth in the conformal space

AbstractWe investigate the asymptotic behavior of global solutions for nonlinear Schrödinger equation with a defocusing exponential nonlinearity in two space dimensions. We prove that any (global) solution approaches a unique free Schrödinger solution at the time infinity in the conformal space. Copyright © 2009 John Wiley & Sons, Ltd.

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ℝ n , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ℝ n . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.

Hot spots for the heat equation with a rapidly decaying negative potential

We consider the Cauchy problem of the heat equation with a radially symmetric, negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$, for some $\kappa>2$, and study the relation between the large-time behavior of hot spots of the solutions and the behavior of the potential at the space infinity. In particular, we prove that the hot spots tend to the space infinity as $t\to\infty$ and how their rates depend on whether $V(|\cdot|)\in L^1({\bf R}^N)$ or not.

On viscous Burgers-like equations with linearly growing initial data

The Burgers-like equation is considered: \partial_ t u − Delta u + div G ( u ) = 0 in R^n \times (0,T). In this paper we consider the case that the initial data is not bounded at the space infinity. This paper specifies the growth of nonlinear term as G ( r ) ~ r^ 2 for large r . A typical example is the viscous Burgers equation. Our goal is to solve the initial value problem when the initial data may grow linearly at the space infinity. We shall prove that the problem admits a unique local regular solution.

The Navier-Stokes flow with linearly growing initial velocity in the whole space

In this paper, the uniqueness of the solutions to the Navier-Stokes equations in the whole space is constructed, provided that the velocity grows linearly at infinity. The velocity can be chosen as Mx + u(x) for some constant matrix M and some function u. The perturbation u is taken in some homogeneous Besov spaces, which contain some nondecaying functions at space infinity, typically, some almost periodic functions. It is also proved that a locally-in-time solution exists, when M is essentially skew-symmetric which demonstrates the rotating fluid in 2- or 3-dimension.

Blow-up directions at space infinity for solutions of semilinear heat equations

A blowing up solution of the semilinear heat equation u_t = \Delta u+f(u) with f satisfying lim inf f(u)=u^p > 0 for some p > 1 is considered when initial data u_0 satisfies u_0 \leq M, u\ne M and lim_{m\rightarrow infity} inf_{x\in B_m} u_0(x) = M with sequence of ball {B_m} whose radius diverging to infinity. It is shown that the solution blows up only at space infinity. A notion of blow-up direction is introduced. A characterization for blow-up direction is also established.

Correction to "Blow-up directions at space infinity for solutions of semilinear heat equations" BSPM 23 (2005), 9-28.

This work presents corrections to Blow-up directions at space infnity for solutions of semilinear heat equations published in BSPM 23(2005), 9-28.

INFINITY ON THE THRESHOLD OF CHRISTIANITY: THE EMERGENCE OF A POSITIVE CONCEPT OUT OF NEGATIVITY

1. Introduction The repudiation of infinity, or horror infiniti, has generally been considered as an insignia of ancient culture. (For a general introduction to the topic, see e.g. Moore 1993, Moore 1990, Maor 1987, de Vogel 1959, and Cohn 1896.) In the same vein it is usually argued that a significant change occurred in the attitude to infinity at the beginning of the Christian era and its newborn religion. As Christians made an attempt to buttress their revelatory experience of God also on the level of rationality, relying in this action on the rich philosophical heritage of the ancients, there was introduced a kind of clash between finitist and infinitist elements into the thought structure of the Christian world. I would like to argue that this substantial conflict provides us with contextual means for investigating not only the concept of theological infinity, elaborated in the Middle Ages, but enables us to look in a historically consecutive way also at the two modern children of infinity - at the idea of physical infinity, starting to emerge in the Renaissance, and at aesthetic infinity, appearing in the second half of the 18th century. In other words, I claim that the history of infinity offers us scope by which to analyze the mutually instigating relationship among faith, knowledge and artistic representation: it reveals how a certain idea, accepted on the grounds of religious arguments, can force the modification of the speculative system of the age and lead, in terms of the modern era, to the foundation of new scientific (experimental) truths, as well as of a new system of artistic representation. Keeping in sight such a project of cultural rhetoric of infinity, I will confine myself in the present article to the topic of the revaluation of infinity in late antiquity and early Christianity. 2. Platonic-Aristotelian concept of infinity Before going on to tackle the proper subject of the paper, it is necessary to offer some explanatory remarks about the Greeks' disgust of infinity. First, we should make it clearer what we mean by the word infinity because its usual Greek equivalent, apeiron, had a range of meanings not necessarily identical with our sense of the word (Guthrie 1962:83-89, Sweeney 1992:15-28). For example, it is questionable how much we can attribute to it the strictly spatial connotations prevalent in our perception. Aristotle had stated that the belief in the existence of infinity can be motivated in us by five different impressions producing the respective modes of infinity: temporal eternity, infinite divisibility of quantities, the endless process of generation and perishing, the absence of a final boundary line in the corporeal world, the additive infinity in numbers and in our thought (Physics 203b16-269); some steps further Aristotle comes to conclude that infinity as such can exist either by way of addition or by way of division (Physics 204a6-8). As to Aristotle's personal conception of infinity, he coupled it, as is well known, with potentiality pertinent to matter, admitting, quite interestingly, the possibility of endless division as regards spatial quantity, while denying the infinity in space by means of addition (which, i.e. additive infinity, he was in turn ready to admit in respect to time) (Moore 1990: 40-41). In spite of the subtleties we have to consider in speaking about apeiron in various Greek philosophers, I would wish for the moment to stay on the more general plane and to declare simply that the so-called Greek abhorrence of infinity refers to the lowest position infinity as indefiniteness necessarily held, as compared to all finitist conceptions, in traditional Greek thought. In other words, not before Christianity and Neoplatonism (Plotinus), was infinity assigned in the Greek mind a significant cognitive value, let alone elevating it to the position of highest perfection. Nevertheless, even this broad statement needs some qualifications and can be maintained only with the proviso that we limit our scope of vision chiefly to the mainstream of ancient philosophy, i. …

Explosive behavior in spatially discrete reaction-diffusion systems

Explosive instabilities in spatially discrete reaction-diffusion systems are studied. We identify classes of initial data developing singularities in finite time and obtain predictions of the blow-up times, whose accuracy is checked by comparison with numerical solutions. We present averaged and local blow-up estimates. Local blow-up results show that it is possible to have blow-up after blow-up. Conditions excluding or implying blow-up at space infinity are discussed.

Random attractors for the stochastic Benjamin–Bona–Mahony equation on unbounded domains

We prove the existence of a compact random attractor for the stochastic Benjamin–Bona–Mahony equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.

Spacelike hypersurfaces with constant mean curvature in the steady state space

We consider complete spacelike hypersurfaces with constant mean curvature in the open region of de Sitter space known as the steady state space. We prove that if the hypersurface is bounded away from the infinity of the ambient space, then the mean curvature must be H = 1. Moreover, in the 2-dimensional case we obtain that the only complete spacelike surfaces with constant mean curvature which are bounded away from the infinity are the totally umbilical flat surfaces. We also derive some other consequences for hypersurfaces which are bounded away from the future infinity. Finally, using an isometrically equivalent model for the steady state space, we extend our results to a wider family of spacetimes.

Large time behaviors of hot spots for the heat equation with a potential

We consider the Cauchy problem of the heat equation with a potential which behaves like the inverse square at infinity. In this paper we study the large time behavior of hot spots of the solutions for the Cauchy problem, by using the asymptotic behavior of the potential at the space infinity.

Publisher’s Note: Influence of mass multipole moments on the deflection of a light ray by an isolated axisymmetric body [Phys. Rev. D77, 044029 (2008)

Future space astrometry missions are planned to measure positions and/or parallaxes of celestial objects with an accuracy of the order of the microarcsecond. At such a level of accuracy, it will be indispensable to take into account the influence of the mass multipole structure of the giant planets on the bending of light rays. Within the parametrized post-Newtonian formalism, we present an algorithmic procedure enabling to determine explicitly this influence on a light ray connecting two points located at a finite distance. Then we specialize our formulae in the cases where 1) the light source is located at space infinity, 2) both the light source and the observer are located at space infinity. We examine in detail the cases where the unperturbed ray is in the equatorial plane or in a meridian plane.

Influence of mass multipole moments on the deflection of a light ray by an isolated axisymmetric body

Future space astrometry missions are planned to measure positions and/or parallaxes of celestial objects with an accuracy of the order of the microarcsecond. At such a level of accuracy, it will be indispensable to take into account the influence of the mass multipole structure of the giant planets on the bending of light rays. Within the parametrized post-Newtonian formalism, we present an algorithmic procedure enabling us to determine explicitly this influence on a light ray connecting two points located at a finite distance. Then we specialize our formulas in the cases where (1) the light source is located at space infinity, (2) both the light source and the observer are located at space infinity. We examine in detail the cases where the unperturbed ray is in the equatorial plane or in a meridian plane.0.

Asymptotic analysis of the p-Laplacian flow in an exterior domain

We consider the Dirichlet problem for the p-Laplacian evolution equation, ut=Δpu, where p>2, posed in an exterior domain in RN, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p. We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p, which is mathematically more interesting. In this first paper we treat only the cases N>p and N=p, the case N<p will be considered in a companion work.

The global profile of blow-up at space infinity in semilinear heat equations

We consider semilinear heat equations on ${\mathbb{R}}^N$ and discuss the blow-up of solutions that occurs only at space infinity. We give sufficient conditions for such phenomena, and study the global profile of solutions at the blow-up time. Among other things, we establish a nearly optimal upper bound for the blow-up profile, which shows that the profile $u(x,T)$ cannot grow too fast as $|x|\to \infty$. We also prove that such blow-up is always complete.

On Instant Blow-up for Semilinear Heat Equations with Growing Initial Data

For a semilinear heat equation admitting blow-up solutions a sufficient condition for nonexistence of local-in-time solutions are obtained. In particular, it is shown that if an initial data tends to infinity at space infinity then there is no local-in-time solution. As an application if the solution blows up at space infinity with least blow-up time, the solution cannot be extendable after blow-up time.

A Uniform Quantum Version of the Cherry Theorem

Consider in \(L^2(\mathbb{R}^2)\) the operator family \(H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0\) . P0 is the quantum harmonic oscillator with diophantine frequency vector ω, F0 a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and \(\epsilon \in {\mathbb{C}}\) . Then there exist \(\epsilon^\ast > 0\) independent of \(\hbar\) and an open set \(\Gamma\subset{\mathbb{C}}^2\setminus{\mathbb{R}}^2\) such that if \(|\epsilon| < \epsilon^\ast\) and \(\omega\in\Gamma\) , the quantum normal form near P0 converges uniformly with respect to \(\hbar\) . This yields an exact quantization formula for the eigenvalues, and for \(\hbar=0\) the classical Cherry theorem on convergence of Birkhoff’s normal form for complex frequencies is recovered.

On directional blow-up for quasilinear parabolic equations with fast diffusion

We discuss blow-up at space infinity of solutions to quasilinear parabolic equations of the form u t = Δ ϕ ( u ) + f ( u ) with initial data u 0 ∈ L ∞ ( R N ) , where ϕ and f are nonnegative functions satisfying ϕ ″ ⩽ 0 and ∫ 1 ∞ d ξ / f ( ξ ) < ∞ . We study nonnegative blow-up solutions whose blow-up times coincide with those of solutions to the O.D.E. v ′ = f ( v ) with initial data ‖ u 0 ‖ L ∞ ( R N ) . We prove that such a solution blows up only at space infinity and possesses blow-up directions and that they are completely characterized by behavior of initial data. Moreover, necessary and sufficient conditions on initial data for blow-up at minimal blow-up time are also investigated.

Application of dressing method for long wave–short wave resonance interaction equation

In this paper we investigate the application of the Zakharov-Shabat dressing method [Funct. Anal. Appl. 8, 226 (1974)] to (2+1)-dimensional long wave–short wave resonance interaction equation (LSRI). Using this method, we can construct the exact N-soliton solution of this equation depending on arbitrary constants. It contains both solutions which do not decay along N different directions in space infinity and “dromion” ones or localized solutions that decay exponentially in all directions.