Asymptotic Behavior Of Energy Research Articles

Concentration profile, energy, and weak limits of radial solutions to semilinear elliptic equations with Trudinger–Moser critical nonlinearities

We investigate the next Trudinger–Moser critical equations, $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda ue^{u^2+\alpha |u|^\beta }&{}\text { in }B, u=0&{}\text { on }\partial B, \end{array}\right. } \end{aligned}$$ where $$\alpha >0$$ , $$(\lambda ,\beta )\in (0,\infty )\times (0,2)$$ and $$B\subset {\mathbb {R}}^2$$ is the unit ball centered at the origin. We classify the asymptotic behavior of energy bounded sequences of radial solutions. Via the blow–up analysis and a scaling technique, we deduce the limit profile, energy, and several asymptotic formulas of concentrating solutions together with precise information of the weak limit. In particular, we obtain a new necessary condition on the amplitude of the weak limit at the concentration point. This gives a proof of the conjecture by Grossi et al. (Math Ann, to appear) in 2020 in the radial case. Moreover, in the case of $$\beta \le 1$$ , we show that any sequence carries at most one bubble. This allows a new proof of the nonexistence of low energy nodal radial solutions for $$(\lambda ,\beta )$$ in a suitable range. Lastly, we discuss several counterparts of our classification result. Especially, we prove the existence of a sequence of solutions which carries multiple bubbles and weakly converges to a sign-changing solution.

Clustering impact regime with shocks in freely evolving granular gas

A freely cooling granular gas without any external force evolves from the initial homogeneous state to the inhomogeneous clustering state, at which the energy decay deviates from the Haff’s law. The asymptotic behavior of energy in the inelastic hard sphere model have been predicted by several theories, which are based on the mode coupling theory or extension of inelastic hard rods gas. In this study, we revisited the clustering regime of freely evolving granular gas via large-scale molecular dynamics simulation with up to 16.7 million inelastic hard disks. We found novel regime regarding on collisions between “clusters” spontaneously appearing after clustering regime, which can only be identified more than a few million particles system. The volumetric dilatation pattern of semicircular shape originated from density shock propagation are well characterized on the appearing of “cluster impact” during the aggregation process of clusters.

Domain Structure of Bulk Ferromagnetic Crystals in Applied Fields Near Saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a branched domain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp-interface energy giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy, and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples.

The energy decay problem for wave equations with nonlinear dissipative terms in $\mathbb{R}^n$

We study the asymptotic behavior of energy for wave equations with nonlinear damping g(u t ) = |u t | m-1 ut in R n (n ≥ 3) as time t → ∞. The main result shows a polynomial decay rate of energy under the condition 1 < m < (n + 2) /(n + 1). Previously, only logarithmic decay rates were found.

Numerical evaluation of the asymptotic energy behavior of intermediate shells with application to two classical benchmark tests

We consider the class of shell problems which are neither purely bending neither purely membrane dominated. In such cases the asymptotic energy norm behavior (which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes) is not a priori known. In this work we apply a numerical procedure in order to estimate the energy behavior of a general shell problem. In order to test its reliability, the method is applied to various problems for which the theoretical energy behavior is known and the results can be compared. Among the problems tested, we have two classical engineering shell benchmarks which are neither bending neither membrane dominated, and for which an analytical evaluation has been obtained in a recent work. All the energy behavior estimates obtained with the numerical method are in perfect agreement with the theoretical values.

Ultrahigh energy neutrino-nucleon interactions

Ultra-high energy ($E_\nu>10^8 $ GeV) neutrino-nucleon total cross sections $\sigma^{\nu N}$ are estimated from a soft non-perturbative model complemented by an approximate QCD evolution, explicitly calculated. The resulting asymptotic energy behavior of the neutrino-nucleon charged-current total cross section is derived analytically and predictions concerning the observation of ultra-high energy neutrinos in future experiments are presented.

Asymptotic Energy Behavior of Two Classical Intermediate Benchmark Shell Problems

We consider two classical problems which are widely used as benchmark tests for shell numerical methods: the Scordelis–Lo roof and the pinched roof. Due to the particular load and boundary conditions applied, neither belongs to the well-known classes of purely bending or purely membrane dominated shells. Consequently the asymptotic energy norm behavior, which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes, is not a priori known. In this work, using space interpolation techniques and a recently developed "intermediate" shell theory, the asymptotic energy behavior of both problems is found analytically. The results are in agreement with the numerical estimates obtained in other papers.

Logarithmic supersymmetric electroweak effects on four-fermion processes at TeV energies

We compute the minimal supersymmetry model one-loop contributions to the asymptotic energy behavior of fermion-antifermion pair production at future lepton-antilepton colliders. In addition to the conventional logarithms of renormalization group origin, extra supersymmetry linear logarithmic terms appear of ``Sudakov-type.'' In the TeV range their overall effect on a variety of observables can be quite relevant and drastically different from that obtained in the standard model case.

Role of the top quark mass inbproduction at future lepton colliders

We compute the one loop contribution coming from vertex and box diagrams, where virtual top quarks are exchanged, to the asymptotic energy behavior of $b\overline{b}$ pair production at future lepton colliders. We find that the effect of the top quark mass is an extra linear logarithmic term of Sudakov type that is not present in the case of $(u,d,s,c)$ production. This appears to be particularly relevant in the case of the $b\overline{b}$ cross section.

Asymptotic behaviour of energy radiated by randomly positioned sources of oscillations

The energy radiated by point sources of oscillations having a Poisson distribution is calculated.

Rearranged Perturbation Theory for the Schrödinger Equation

The perturbation theory expansion has been rearranged for the Schrödinger equation using the nonlinear transformation of the perturbation parameter and determining the asymptotic behaviour of energy. It is shown that the use of the low-order Padé-approximants for the rearranged series allows accurate values of energy to be obtained not only for small but also for large values of the perturbation parameter.

Proton-Proton Total Cross Section above104GeV: Can Cosmic Rays Give the Answer?

From Glauber calculations of the absorption cross section of nucleons on air (${\ensuremath{\sigma}}_{\mathrm{abs}}$) we find that present extensive-air-shower data cannot resolve the question of the asymptotic energy behavior of the nucleon-nucleon total cross section [${\ensuremath{\sigma}}_{t}(\mathrm{NN})$]. Relative differences between various asymptotic extrapolations for ${\ensuremath{\sigma}}_{t}(\mathrm{NN})$ are reduced by about a factor of 3 in ${\ensuremath{\sigma}}_{\mathrm{abs}}$ through the Glauber conversion. Novel techniques or conventional experiments with a better determination of electron and muon numbers in extensive air showers will be required in order to obtain useful information on proton-proton cross sections in the asymptotic region.