Infinity In Space Research Articles

Liouville comparison principles for solutions of semilinear elliptic second-order partial differential inequalities

The purpose of this work is to obtain new Liouville comparison principles for entire weak solutions of semilinear elliptic second-order partial differential inequalities of the forms and on ℝ n , where n ≥ 2, q > 0 and ℒ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form, in terms of the behaviour of the coefficients of the operator ℒ at infinity in a function space directly associated with ℒ.

Life span of solutions for a quasilinear parabolic equation with initial data having positive limit inferior at infinity

We present a new upper bound of the life span of positive solutions of a quasilinear parabolic equation for the initial data having positive limit inferior at space infinity. The upper bound is expressed by the data in limit inferior, not in every direction, but around a specific direction. It is also shown that the minimal time blow-up occurs when the initial data attain its maximum at space infinity.

Sharp decay estimates of [formula omitted]-norms for nonnegative Schrödinger heat semigroups

Let H=−Δ+V be a nonnegative Schrödinger operator on L2(RN), where N⩾3 and V is a radially symmetric nonpositive function in RN decaying quadratically at the space infinity. For any 1⩽p⩽q⩽∞, we denote by ‖e−tH‖q,p the operator norm of the Schrödinger heat semigroup e−tH from Lp(RN) to Lq(RN). In this paper, under suitable conditions on V, we give the exact and optimal decay rates of ‖e−tH‖q,p as t→∞ for all 1⩽p⩽q⩽∞. The decay rates of ‖e−tH‖q,p depend on whether the operator H is subcritical or critical and on the behavior of the positive harmonic function for the operator H.

AdS field theory from conformal field theory

Abstract We provide necessary and sufficient conditions for a Conformal Field Theory to have a description in terms of a perturbative Effective Field Theory in AdS. The first two conditions are well-known: the existence of a perturbative ‘1/N ’ expansion and an approximate Fock space of states generated by a finite number of low-dimension operators. We add a third condition, that the Mellin amplitudes of the CFT correlators must be well- approximated by functions that are bounded by a polynomial at infinity in Mellin space, or in other words, that the Mellin amplitudes have an effective theory-type expansion. We explain the relationship between our conditions and unitarity, and provide an analogy with scattering amplitudes that becomes exact in the flat space limit of AdS. The analysis also yields a simple connection between conformal blocks and AdS diagrams, providing a new calculational tool very much in the spirit of the S-Matrix program. We also begin to explore the potential pathologies associated with higher spin fields in AdS by generalizing Weinberg’s soft theorems to AdS/CFT. The AdS analog of Weinberg’s argument constrains the interactions of conserved currents in CFTs, but there are potential loopholes that are unavailable to theories of massless higher spin particles in flat spacetime.

Traveling wave front and stability as planar wave of reaction diffusion equations with nonlocal delays

This paper is concerned with traveling wave front and the stability as planar wave of reaction diffusion system on $${\mathbb{R}^{n}}$$ , where n ≥ 2. Existence and asymptotic behavior of traveling wave front are discussed firstly. The stability as planar wave is established secondly by using super-sub solution method. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar wave as $${t \rightarrow {\infty}}$$ and the convergence is uniform in $${\mathbb{R}^{n}}$$ .

Dynamics for a complex-valued heat equation with an inverse nonlinearity

We study the Cauchy problem for a parabolic system which is derived from a complex-valued heat equation with an inverse nonlinearity. First, we provide some criteria for the global existence of solutions. Then we consider the case when the initial data are asymptotically constants and obtain that, depending on the asymptotic limits, the solution quenches at space infinity or exists globally in time.

The Cauchy problem for a nonhomogeneous heat equation with reaction

We study the behaviour of the solutions to the Cauchy problem$$\left\{\begin{array}{ll}\rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N ,\end{array}\right.$$with $p>0$ and a positive density $\rho$ satisfying$\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0 We show that instantaneous blow-up at space infinity takes placewhen $p\le1$. We also briefly discuss the case $2<\sigma< N$: we prove that theFujita exponent in this case does not depend on $\sigma$,$\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too smallat infinity a phenomenon of instantaneous complete blow-up occurs inthe range $1< p < \tilde{p}_c$

Elastic wave omnidirectional absorbers designed by transformation method

A conformal mapping is proposed to design with transformation method broadband elastic wave omnidirectional absorbers. This conformal mapping transforms the infinity of the virtual space into the origin of the physical space and keeps the transformed material isotropic. The material realization of such elastic wave absorber is proposed and validated by numerical simulation. Different from the existing procedure based on Hamiltonian optics, the proposed method is a more general one and can be applied to any physical process where the transformation method is valid. It also includes the existing techniques as the special cases for electromagnetic and acoustic waves.

Stability of planar waves in the buffered bistable system

This paper is concerned with the asymptotic stability of planar waves in the buffered bistable system on , where n ≥ 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t → ∞ . The convergence is uniform in . Copyright © 2012 John Wiley &amp; Sons, Ltd.

Formula omitted] norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots

This paper is concerned with the Cauchy problem for the heat equation with a potential(P){∂tu=Δu−V(|x|)uin RN×(0,∞),u(x,0)=ϕ(x)in RN, where ∂t=∂/∂t, N⩾3, ϕ∈L2(RN), and V=V(|x|) is a smooth, nonpositive, and radially symmetric function having quadratic decay at the space infinity. In this paper we assume that the Schrödinger operator H=−Δ+V is nonnegative on L2(RN), and give the exact power decay rates of Lq norm (q⩾2) of the solution e−tHϕ of (P) as t→∞. Furthermore we study the large time behavior of the solution of (P) and its hot spots.

Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow

We study a diophantine property for translation surfaces, defined in term of saddle connections and inspired by the classical theorem of Khinchin. We prove that the same dichotomy holds as in Khinchin' result, then we deduce a sharp estimation on how fast the typical Teichmuller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one.

Local quasi-concavity of the solutions of the heat equation with a nonnegative potential

In this paper we consider the Cauchy problem for the heat equation with a nonnegative potential decaying quadratically at the space infinity and investigate local concavity properties of the solution. In particular, we give a sufficient condition for the solution to be quasi-concave in a ball for any sufficiently large t, and discuss the optimality of the sufficient condition, identifying a threshold for the occurrence of local quasi-concavity.

Stability of planar waves in reaction-diffusion system

This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on ℝ n , where n ⩾ 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t → ∞. The convergence is uniform in ℝ n . Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.

Life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity

We present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity. The upper bound is expressed by the data in limit inferior, not in every direction, but around a specific direction. It is also shown that the minimal time blow-up occurs when initial data attains its maximum at space infinity.

Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differentialequation with a power nonlinearity.It was shown in our previous paper that insome parameter range, the problem has a solution with a moving singularitythat becomes anomalous in finite time.Our concern is a blow-up solution with a moving singularity.In this paper, we show that there exists a solution with a moving singularity such thatit blows up at space infinity.

Blow-Up at Space Infinity for Solutions of Cooperative Reaction-Diffusion Systems

We consider the Cauchy problem of cooperative reaction-diffusion systems with nonnegative initial data. Here we discuss the blow-up of a solution that occurs only at space infinity. We give sufficient conditions for such phenomena, and study an asymptotic behaviour at space infinity of the solutions at the blow-up time. In general, relatively little is proved on the locations of blow-up point for semilinear parabolic systems. However, our results can be applied to a large class of nonlinearity for some class of initial value. The reason of this is that, when the blow-up occurs only at space infinity, the effect of reaction is much stronger than that of diffusion, and the behaviour at space infinity is well approximated by the flat solution.

On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow

We study the motion of noncompact hypersurfaces moved by their mean curvature obtained by a rotation around $x$-axis of the graph a function $y=u(x,t)$ (defined for all $x\in \mathbb{R}$). We are interested to estimate its profile when the hypersurface closes open ends at the quenching (pinching) time $T$. We estimate its profile at the quenching time from above and below. We in particular prove that $u(x,T)$ ~ $|x|^{-a}$ as $|x|\to\infty$ if $u(x,0)$ tends to its infimum with algebraic rate $|x|^{-2a} $ (as $|x| \to \infty $ with $a>0$).

A Hilbert–Mumford criterion for polystability in Kaehler geometry

Consider a Hamiltonian action of a compact Lie group K on a Kaehler manifold X with moment map μ: X → * . Assume that the action of K extends to a holomorphic action of the complexification G of K. We characterize which G-orbits in X intersect μ ―1 (0) in terms of the maximal weights lim t→∞ 〈μ(e its · x), s), where s ∈ . We do not impose any a priori restriction on the stabilizer of x. Under some mild restrictions on the action K X, we view the maximal weights as defining a collection of maps: for each x ∈ X, λ x : ∂ ∞ (K\G) → ℝ ∪ {∞}, where ∂ ∞ (K\G) is the boundary at infinity of the symmetric space K\G. We prove that G · x ∩ μ ―1 (0) ≠ 0 if: (1) λ x is everywhere nonnegative, (2) any boundary point y such that λ x (y) = 0 can be connected with a geodesic in K\G to another boundary point y' satisfying λ x (y') = 0. We also prove that the maximal weight functions are G-equivariant: for any g ∈ G and any y ∈ ∂ ∞ (K\G) we have λ g·x (y) = λ x (y · g).

BLOW-UP OF SMOOTH SOLUTIONS TO THE NAVIER–STOKES EQUATIONS OF COMPRESSIBLE VISCOUS HEAT-CONDUCTING FLUIDS

AbstractWe give a simpler and refined proof of some blow-up results of smooth solutions to the Cauchy problem for the Navier–Stokes equations of compressible, viscous and heat-conducting fluids in arbitrary space dimensions. Our main results reveal that smooth solutions with compactly supported initial density will blow up in finite time, and that if the initial density decays at infinity in space, then there is no global solution for which the velocity decays as the reciprocal of the elapsed time.

Asymptotic Behavior for a Class of Modified -Potentials in a Half Space

A class of -potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates at infinity in the upper-half space of the -dimensional Euclidean space, where the function is a positive non-decreasing function on the interval satisfying certain conditions. This result generalizes the growth properties of analytic functions, harmonic functions, and superharmonic functions.