Time Asymptotic Behavior Research Articles

Generalized volume-complexity for Lovelock black holes

We study the time dependence of the generalized complexity of Lovelock black holes using the “complexity = anything” conjecture, which expands upon the notion of “complexity = volume” and generates a large class of observables. By applying a specific condition, a more limited class can be chosen, whose time growth is equivalent to a conserved momentum. Specifically, we investigate the numerical full time behavior of complexity time rate, focusing on the second and third orders of Lovelock theory coupled with Maxwell term, incorporating an additional term – the square of the Weyl tensor of the background spacetime – into the generalization function. Furthermore, we repeat the analysis for case with three additional scalar terms: the square of Riemann and Ricci tensors, and the Ricci scalar for second-order gravity (Gauss–Bonnet) showing how these terms can affect to multiple asymptotic behavior of time. We study how the phase transition of generalized complexity and its time evolution occur at turning point (τturning) where the maximal generalized volume supersedes another branch. Additionally, we discuss the late time behavior, focusing on proportionality of the complexity time rate to the difference of temperature times entropy at the two horizons (TS(r+)-TS(r-)) for charged black holes, which can be corrected by generalization function of each radius in generalized case. In this limit, we also explore near singularity structure by approximating spacetime to Kasner metrics and finding possible values of complexity growth rate with different choices of the generalization function.

On the decay of one-dimensional locally and partially dissipated hyperbolic systems

We study the time-asymptotic behavior of linear hyperbolic systems subject to partial dissipation that is localized in suitable subsets of the domain. Specifically, we recover the classical decay rates of partially dissipative systems that satisfy the stability condition (SK), with a time-delay that depends only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space interval and that the system, without space-restriction on the dissipation, satisfies the stability condition (SK). The former assumption ensures that the time spent by the characteristics of the system in the undamped region is finite, and the latter ensures that the solutions decay whenever the damping is active. Our approach consists of reformulating the system into $n$ coupled transport equations and showing that the time-decay estimates are delayed by the sum of the times that each characteristic spends in the undamped region.

Asymptotic behavior in time of a generalized Navier-Stokes-alpha model

Asymptotic behavior in time of a generalized Navier-Stokes-alpha model

Stability and receptivity analyses of the heated flat-plate boundary layer with variable viscosity

Stability and receptivity analyses of the heated flat-plate boundary layer with variable viscosity

Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion

Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion

Integrability and Inverse Scattering Transform of the Modified Benjamin-Ono Equation

Abstract This paper presents a B{"a}cklund transformation, the Lax representation, and conserved quantities for the modified Benjamin--Ono equation. The initial problem of the modified Benjamin--Ono equation on the line was studied by the inverse scattering transform method, presenting a nonlocal Riemann--Hilbert problem in the inverse problem to reconstruct the explicit potential function. Further, the exact $N$-soliton solutions and long--time asymptotic behavior are provided, respectively. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms. As a revised version, the mBO equation incorporates the solution features of the BO equation, and its solutions are presented in logarithmic form, which seeks to describe related physical phenomena more accurately. Our results will contribute to further exploration of physical and mathematical problems related to the mBO equation.

On the Cauchy problem for acoustic waves in hereditary fluids: Decay properties and inviscid limits

This paper considers the viscous/inviscid Moore–Gibson–Thompson (MGT) equations with memory of type I in the whole space . For one thing, associating with a new condition on initial data, we derive the optimal estimates and the optimal leading term of the acoustic velocity potential for large time, where we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, by using the multiscale analysis and energy methods in the Fourier space, we demonstrate the inviscid limits (i.e., as the diffusivity of sound tends to zero), which match our Wentzel‐Kramers‐Brillouin (WKB) expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan‐MGT equation in viscous hereditary fluids.

Wave propagation in three-dimensional fractional viscoelastic infinite solid body

Wave propagation in three-dimensional fractional viscoelastic infinite solid body

On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup

<p>This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.</p>

Time-asymptotic behavior of solution for NLS with large-size initial value

Time-asymptotic behavior of solution for NLS with large-size initial value

Population growth in discrete time: a renewal equation oriented survey

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.

The Hardy inequality and large time behaviour of the heat equation on [formula omitted

The Hardy inequality and large time behaviour of the heat equation on [formula omitted

Dynamics of a spatially homogeneous Vicsek model for oriented particles on a plane

We consider a spatially homogeneous Kolmogorov–Vicsek model in two dimensions, which describes the alignment dynamics of self-driven stochastic particles that move on a plane at a constant speed, under space-homogeneity. In [A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of spatially homogeneous Kolmogorov–Vicsek model as a gradient flow, Arch. Rational Mech. Anal. 227 (2018) 869–896] Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior is obtained for the two-dimensional case, due to the failure of the celebrated Bakery and Emery condition for the logarithmic Sobolev inequality. We prove exponential convergence (with quantitative rate) of the weak solutions towards a Fisher-von Mises distribution, using a new condition for the logarithmic Sobolev inequality.

On echoes in magnetohydrodynamics with magnetic dissipation

We study the long time asymptotic behavior of the inviscid magnetohydrodynamic equations with magnetic dissipation near a combination of Couette flow and a constant magnetic field. Here we show that there exist nearby explicit global in time low frequency solutions, which we call waves. Moreover, the linearized problem around these waves exhibits resonances under high frequency perturbations, called echoes, which result in norm inflation in Gevrey regularity and infinite time blow-up in Sobolev regularity.

Modified Scattering for the One-Dimensional Schrödinger Equation with a Subcritical Dissipative Nonlinearity

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity $$\lambda |u|^\alpha u$$ , where $$0<\alpha <2$$ , and $$\lambda $$ is a complex constant satisfying $$\text {Im} \lambda >\frac{\alpha |\text {Re} \lambda |}{2\sqrt{ \alpha +1}}$$ . For arbitrary large initial data, we present the uniform time decay estimates when $$4/3\le \alpha <2$$ , and the large time asymptotics of the solution when $$\frac{7+\sqrt{145}}{12}<\alpha <2$$ . The proof is based on the vector fields method and a semiclassical analysis method.

A stochastic predator–prey model with Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction and probability density function

A stochastic predator–prey model with Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction and probability density function

A small time large deviation principle for stochastic differential delay equations

A small time large deviation principle for stochastic differential delay equations

Modified scattering for the derivative fractional nonlinear Schrödinger equation

Modified scattering for the derivative fractional nonlinear Schrödinger equation

Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws

We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.

Time evolution of a decaying quantum state: Evaluation and onset of nonexponential decay

The method developed by van Dijk, Nogami, and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of noncompact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of nonexponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the half-line.