Arbitrary Initial Energy Research Articles

Generality of Starobinsky and Higgs inflation in the Jordan frame

We revisit the problem of generality of Starobinsky and Higgs inflation. The known results obtained in the Einstein frame are generalized for the case of an arbitrary initial energy of the scalar field. These results are compared with the results obtained directly in the Jordan frame, which, to our knowledge, has not been thoroughly explored in the literature previously. We demonstrate that the qualitative picture of initial conditions zone in the $(\phi, \dot \phi)$ plane, which leads to sufficient amount of inflation, is quite similar for both the frames in the case of Higgs inflation. For Starobinsky inflation, the conformal transformation between the frames relates the geometrical variables in the Jordan frame with the properties of an effective scalar field in the Einstein frame. We show that the transformation $(H, R) \to (\phi, \dot \phi)$ is not regular everywhere, leading to some peculiarities in the zone of good initial conditions in the $(H, R)$ plane.

Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity $$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+(-\Delta )^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} {\mathbb {R}}^{N}\backslash \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N \, ( N\ge 1)$$ is a bounded domain with Lipschitz boundary and $$2\le p< \infty $$ . The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.

Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity

&lt;p style='text-indent:20px;'&gt;In this paper, an initial boundary value problem for a parabolic type Kirchhoff equation with time-dependent nonlinearity is considered. A new blow-up criterion for nonnegative initial energy is given and upper and lower bounds for the blow-up time are also derived. These results partially generalize some recent ones obtained by Han and Li in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(2018), 3283-3297].&lt;/p&gt;

Global Existence and Blow-Up for a Kirchhoff-Type Hyperbolic Problem with Logarithmic Nonlinearity

This paper deals with a Kirchhoff-type hyperbolic problem with logarithmic nonlinearity. Firstly, by employing the classical Galerkin method and potential well method, we show the existence of the global weak solution with subcritical initial energy and critical initial energy. Secondly, we obtain the finite time blow up results with subcritical initial energy and critical initial energy by using the concavity method. In addition, under some suitable conditions, we also estimate the upper and lower bounds of the blow-up time for the blow-up solution by using the differential inequality technique. Finally, we establish a new finite time blow up condition which is independent of the depth of the potential well and we obtain the upper bound of the blow-up time for the blow-up solution. Furthermore, we also prove the weak solution can blow up in finite time at arbitrary initial energy level by using this condition.

Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy

<p style='text-indent:20px;'>The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}-\Delta u-\Delta u_{t} = |u|^{p-2}u\log|u|, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which has been studied in [<xref ref-type="bibr" rid="b5">5</xref>]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the <inline-formula><tex-math id="M1">\begin{document}$ L^{2}(\Omega) $\end{document}</tex-math></inline-formula> norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [<xref ref-type="bibr" rid="b5">5</xref>].

Finite time blow-up for some nonlinear evolution equations

Nonexistence of global solutions to a class of high-order nonlinear evolution equations is investigated. Indeed, under sufficient conditions on the datum and the source term, solutions with arbitrary initial energy blow-up in a finite time.

Blow-Up of Solutions to Semilinear Nonautonomous Wave Equations with Robin Boundary Conditions

The problem of the blow-up of solutions to the initial boundary value problem for a nonautonomous semilinear wave equation with damping and accelerating terms under the Robin boundary condition is studied. Sufficient conditions for the blow up in finite time of solutions to semilinear damped wave equations with arbitrary large initial energy are obtained. A result on the blow-up of solutions with negative initial energy to a semilinear second-order wave equation with an accelerating term is also obtained.

Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) &lt; d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) &gt; 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.

Blow-Up Phenomena for a Class of Generalized Double Dispersion Equations

In this article, we study the blow-up phenomena of generalized double dispersion equations utt − uxx − uxxt + uxxxx − uxxtt = f(ux)x. Under suitable conditions on the initial data, we first establish a blow-up result for the solutions with arbitrary high initial energy, and give some upper bounds for blow-up time T* depending on sign and size of initial energy E(0). Furthermore, a lower bound for blow-up time T* is determined by means of a differential inequality argument when blow-up occurs.

GLOBAL EXISTENCE AND FINITE TIME BLOW-UP OF SOLUTIONS TO A NONLOCAL P-LAPLACE EQUATION

In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].

Delay Minimization for Data Transmission in Wireless Power Transfer Systems

Radiative wireless power transfer (WPT) is a promising technique to power wireless devices' transmission. In a resource-limited device, receiving energy and transmitting data cannot operate at the same time because they share the same spectrum or hardware. This paper studies the problem for a wireless device to decide when to harvest energy, when to deliver data, and what transmission rate to use. Distinct from the most existing works, we focus on delay minimization in transmitting a sequence of data packets over a point-to-point channel, which is critical for time-sensitive applications. Since the battery is capacitated, the device must repeatedly switch between harvesting energy and transmitting data. For the offline case where packet information is known before scheduling, a surprising result is discovered that for all (energy receiving and data transmitting) cycles, except the last one, the optimal transmission rate should be a constant which is called the wOPT rate. Based on this discovery, the offline delay minimization problem is optimally solved. For the online case where packets arrive dynamically without prior information, we propose a simple online algorithm: using the wOPT rate to transmit whenever both energy and data are ready. It is proved to be 1.16-competitive if the battery is initially empty, namely, its delay is less than 1.16 times the offline optimal delay for any given packet set. When the battery is with arbitrary initial energy, simulation results show that the performance is near optimal. The discovery of the wOPT rate reveals an essential property of WPT and is expected to be significant in solving other related problems.

Global Existence, Finite Time Blow-up and Vacuum Isolating Phenomena for Semilinear Parabolic Equation with Conical Degeneration

This paper is devoted to studying a semilinear parabolic equation with conical degeneration. First, we extend previous results on the vacuum isolating of solution with initial energy $J(u_0) \lt d$, where $d$ is the mountain pass level. Concretely, we obtain the explicit vacuum region, the global existence region and the blow-up region. Moreover, as far as the blow-up solution is concerned, we estimate the upper bound of the blow-up time and blow-up rate. Second, for all $p \gt 1$, we get a new sufficient condition, which demonstrates the finite time blow-up for arbitrary initial energy, and the upper bound estimate of blow-up time is obtained.

Asymptotic behavior for a fourth-order parabolic equation involving the Hessian

This paper deals with a fourth-order parabolic PDE arising in the theory of epitaxial growth of crystal. For the stationary problem, we find a ground-state solution on its corresponding Nehari manifold by Lagrange multiplier method. As far as the evolution problem is concerned, we study the dynamics for both the global solution and the blow-up solution. Especially, for the global solution, we prove that the energy functional decays exponentially and we get the concretely decay rate. For the blow-up solution, the exponential growth of the solution is obtained, and we also obtain the growth rate. The behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. Moreover, we prove that there exists blow-up solution with arbitrary high initial energy. Finally, for the low initial energy case, we give some equivalent conditions for the solutions blow up in finite time and exist globally, respectively. Our results extend the results got by Escudero et al. (Eur J Appl Math 24(3):437–453, 2013; J Differ Equ 254(6):2515–2531, 2013; Math Method Appl Sci 37(6):793–807, 2014; J Math Pures Appl 103(4):924–957, 2015).

Mathematical behavior of the solutions of a class of hyperbolic-type equation

In this paper, we consider hyperbolic-type equations with initial and Dirichlet boundary conditions in a bounded domain. Under some suitable assumptions on the initial data and source term, we obtain nonexistence of global solutions for arbitrary initial energy.

Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation \t\t\tut−△ut−△u−div(|∇u|2q∇u)=up\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{t}-\\triangle u_{t}-\\triangle u-\\operatorname{div}\\bigl(| \\nabla u|^{2q}\\nabla u\\bigr)=u^{p} $$\\end{document} which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016), where the blow-up phenomena and the lifespan for the initial energy J(u_{0})<0 were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J(u_{0})<0.

Optical torque on a two-level system near a strongly nonreciprocal medium

We investigate the quantum optical torque on an atom interacting with an inhomogeneous electromagnetic environment described by the most general linear constitutive relations. The atom is modeled as a two-level system prepared in an arbitrary initial energy state. Using the Heisenberg equation of motion (HEM) and under the Markov approximation, we show that the optical torque has a resonant and non-resonant part, associated respectively with a spontaneous-emission process and Casimir-type interactions with the quantum vacuum, which can both be written explicitly in terms of the system Green function. Our formulation is valid for any inhomogeneous, dissipative, dispersive, nonreciprocal, and bianisotropic structure. We apply this general theory to a scenario in which the atom interacts with a material characterized by strong nonreciprocity and modal unidirectionality. In this case, the main decay channel of the atom energy is represented by the unidirectional surface waves launched at the nonreciprocal material-vacuum interface. To provide relevant physical insight into the role of these unidirectional surface waves in the emergence of non-trivial optical torque, we derive closed-form expressions for the induced torque under the quasi-static approximation. Finally, we investigate the equilibrium states of the atom polarization, along which the atom spontaneously tends to align due to the action of the torque. Our theoretical predictions may be experimentally tested with cold Rydberg atoms and superconducting qubits near a nonreciprocal material. We believe that our general theory may find broad application in the context of nano-mechanical and bio-mechanical systems.

Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy

In this paper the authors deal with a class of parabolic type Kirchhoff equations, which were considered in Han and Li (2018), where global existence and finite time blow-up of solutions were studied when the initial energy was subcritical, critical and supercritical. Their results are complemented in this paper in the sense that a new blow-up criterion will be given for nonnegative initial energy and upper and lower bounds for blow-up time will be derived.

Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions

In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.

Global existence and blow-up of solutions to a nonlocal parabolic equation with singular potential

In this paper, we study a nonlocal parabolic equation with singular potential on a bounded smooth domain with homogeneous Neumann boundary condition. Firstly, we find a threshold of global existence and blow-up to the solutions of the problem when the initial data is at the low energy level, i.e., J(u0)≤d, where J(u0) is the initial energy and d is the mountain-pass level. Moreover, when J(u0)<d, the vacuum isolating behavior of the solutions is also discussed. Secondly, we prove that there exist solutions of the problem with arbitrary initial energy that blow up in finite time. We also obtain the upper bounds of the blow-up time for blow-up solutions.

A class of fourth-order parabolic equation with arbitrary initial energy

In this paper we apply the modified potential well method to the study of the long time behaviors of solutions to a class of fourth-order parabolic equation in a bounded smooth domain of Rn for arbitrary n≥1. Global existence and blow up in finite time of solutions are obtained when the initial data satisfy different conditions. To be a little more precise, we give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. Moreover, the decay rate of the L2 norm is also obtained for global solutions. Sufficient conditions for the existence of global and blow-up solutions are also provided for supercritical initial energy. These improve and generalize some recent results.