Low Initial Energy Research Articles

Global Existence and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Pseudo-parabolic Equations

In this paper, we study the initial-boundary value problem for the semilinear pseudo-parabolic equations ut − ΔXut − ΔXu = |u|p−1u, where X = (X1,X2,…,Xm) is a system of real smooth vector fields which satisfy the Hormander’s condition, and ΔX = $${{\rm{\Delta}}_X} = \sum\nolimits_{j = 1}^m {X_j^2} $$ is a finitely degenerate elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.

Global Existence, Finite Time Blow-Up, and Vacuum Isolating Phenomenon for a Class of Thin-Film Equation

We consider a thin-film equation modeling the epitaxial growth of nanoscale thin films. By exploiting the boundary conditions and the variational structure of the equation, we look for conditions on initial data that ensure the solution exists globally or blows up in finite time. Moreover, for global solution, we establish the exponential decays of solutions and energy functional, and give the concrete decay rate. As for blow-up solution, we prove that the solution grows exponentially and obtain the behavior of energy functional as t tends to the maximal existence time. Under the low initial energy, we get further two necessary and sufficient conditions for the solution existing globally and blowing up in finite time, respectively. A new sufficient condition such that the solution exists globally is obtained; we point out that this initial condition is independent to initial energy. Finally, we discuss the vacuum isolating phenomena of the solution.

Integrated modeling of plasma-dielectric interaction: kinetic boundary effects

Kinetic effects of plasma-dielectric interaction are studied theoretically with respect to the mechanisms of electron extraction from solids in response to ion and electron bombardment, coupled with plasma dynamics of a weakly emissive sheath. Emission coefficients of incident beams are first calculated by quantum mechanical as well as semi-classical approaches involving Auger neutralization, energy-dependent ejection due to primary beam, and reflection of low-energy electrons, which are then incorporated into a 1D1V simulation and plasma kinetic theory. Presheath with sheath structures are derived using fluid and kinetic theory regarding ion-induced emission, respectively. The Bohm criterion considering surface emission is evaluated as well. For electron-induced emission, it is found that sheath potential is no longer collinear with plasma electron temperature in our integrated model. Additionally, reflection of low-energy electrons is justified to have a minor impact on the floating sheath for low temperature half-bounded plasma under low pressure. The combined effects of both incident ions and electrons in bounded plasma are analyzed with symmetrical/asymmetrical emission yields at two boundaries. It is then proved that wall potential is barely affected by bulk plasma influx if emission coefficients are symmetrical, while emission due to transiting beam can drastically modify the sheath solution. Electron reflection also becomes more influential if secondary electrons have low initial energy. Finally, we summarize different roles of ion and electron in sheath structure. It is shown that ion-induced emission mitigates sheath potential but cannot reach critical emission on contrast to that of electron flux.

On potential wells to a semilinear hyperbolic equation with damping and conical singularity

This article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by Alimohammady et al. (2017) [1], where the weak solution with low initial energy (I(0)<d,d is the potential well depth) was considered. We extend the previous results in the following three aspects. Firstly, we consider the vacuum isolating behavior of the solution for initial energy I(0)≤0 and 0<I(0)<d, respectively. We find that there are two explicit vacuum regions: an annulus and a ball. Moreover, we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time, and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained. Secondly, we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results. Finally, the weak solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the weak solution exists globally and blows up in finite time are given.

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial energy: the low initial energy case, critical initial energy case and high initial energy case. For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. In addition, for the high initial energy case the possibility of both global existence and finite time blowup is proved first, and then some sufficient initial conditions of finite time blowup and global existence are obtained, respectively.

High energy positron source driven by laser pulses

High energy positron source has many important applications in scientific research. Here we come up with a new internal positron injection scenario that is injecting radionuclides which can emit positrons in β+-decay into laser field. Compared with existing external positron injection scenario, our internal positron injection scenario has advantages in operating and cost. By simulation calculating, we demonstrate the positrons with low initial kinetic energy emitted by radionuclides can be accelerated to GeV by laser pulses.

Microstructural evolution and its influence on toughness in simulated inter-critical heat affected zone of large thickness bainitic steel

Simulated inter-critical heat affected zones (ICHAZ) of a large thickness bainitic steel were prepared with three different peak temperatures (Tp) of 750 °C, 780 °C and 800 °C. The microstructure and crystallographic feature were investigated. Toughness of simulated ICHAZ specimens was assessed using instrumental Charpy impact test at − 40 °C. It was found that the volume fraction of newly formed bainite increased and the effective grain size decreased with the increase in inter-critical Tp. The toughness decreased slightly with the Tp of 750 °C compared with the base metal (BM), while the toughness improved sharply when the Tp exceeded 780 °C. The network-like martensite was primarily responsible for the low crack initiation energy with the Tp of 750 °C. In contrast, the fine bainite formed with the Tp above 780 °C was effective in increasing the crack nucleation resistance, resulting in the high initiation energy. The matrix softening as well as the grain refinement played important roles in improving the crack propagation energy. Furthermore, it was confirmed that the presence of a multi-phase microstructure can lead to the scattering of toughness.

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).

Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity \begin{document}$u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$\end{document} , where \begin{document}$X=(X_1, X_2, ··· , X_m)$\end{document} is an infinitely degenerate system of vector fields, and \begin{document}$\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$\end{document} is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincare inequality, we obtain the global existence and blow-up at \begin{document}$+∞$\end{document} of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

Effect of Deposition Energy on Underwater Electrical Wire Explosion

Underwater electrical wire explosion (UEWE) of the copper wire of 0.3 mm in diameter and 50 mm in length at high and low initial stored energies was compared. For this wire, the completely vaporization energy is 194 J. While the UEWE at high initial stored energy of 961 J show a good reproducibility, the shock waves generated by UEWE at low initial stored energy of 120 J have a poor reproducibility that is attributed to the partial vaporization of the exploding wire. As the initial stored energy was gradually increased from 110 to 1000 J, the deposition energy in the exploding wire was increased from 92 to 246 J. When the deposition energy is higher than the complete vaporization energy of the wire, the relative error in the shock wave is significantly reduced to 4%. The effect of the wire diameter on the UEWEs was investigated by keeping the initial stored energy at 120 J. It was observed that there exist two discharge modes, the mode of the cutting-off current with a partial vaporization for the wires of a diameter smaller than 0.5 mm and the mode of the continuous discharge with no vaporization for the wire of 0.5 mm in diameter. It was found that a partial vaporization of a vaporization rate as low as 6% is enough to generate the shock wave and the amplitude of the shock wave is determined by the vaporized mass of the wire. The more the vaporized mass, the higher is the shock wave.

Asymptotic behavior and blow-up of solutions for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity ut−△Xu=ulog⁡|u|, where X=(X1,X2,⋯,Xm) is an infinitely degenerate system of vector fields, and △X:=∑j=1mXj2 is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the logarithmic Sobolev inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and we also discuss the asymptotic behavior of the solutions.

Optimal perturbation for two-dimensional vortex systems: route to non-axisymmetric state

We investigate perturbations that maximize the gain of disturbance energy in a two-dimensional isolated vortex and a counter-rotating vortex pair. The optimization is carried out using the method of Lagrange multipliers. For low initial energy of the perturbation ( $E(0)$ ), the nonlinear optimal perturbation/gain is found to be the same as the linear optimal perturbation/gain. Beyond a certain threshold $E(0)$ , the optimal perturbation/gain obtained from linear and nonlinear computations are different. There exists a range of $E(0)$ for which the nonlinear optimal gain is higher than the linear optimal gain. For an isolated vortex, the higher value of nonlinear optimal gain is attributed to interaction among different azimuthal components, which is otherwise absent in a linearized system. Spiral dislocations are found in the nonlinear optimal perturbation at the radial location where the most dominant wavenumber changes. Long-time nonlinear evolution of linear and nonlinear optimal perturbations is studied. The evolution shows that, after the initial increment of perturbation energy, the vortex attains a quasi-steady state where the mean perturbation energy decreases on a slow time scale. The quasi-steady vortex state is non-axisymmetric and its shape depends on the initial perturbation. It is observed that the lifetime of a quasi-steady vortex state obtained using the nonlinear optimal perturbation is longer than that obtained using the linear optimal perturbation. For a counter-rotating vortex pair, the mechanism that maximizes the energy gain is found to be similar to that of the isolated vortex. Within the linear framework, the optimal perturbation for a vortex pair can be either symmetric or antisymmetric, whereas the structure of the nonlinear optimal perturbation, beyond the threshold $E(0)$ , is always asymmetric. No quasi-steady state for a counter-rotating vortex pair is observed.

Global existence and blow up of the solution for nonlinear Klein–Gordon equation with general power-type nonlinearities at three initial energy levels

This paper investigates the local existence, global existence and finite time blow up of the solution to the Cauchy problem for a class of nonlinear Klein–Gordon equation with general power-type nonlinearities. We give some sufficient conditions on the initial data such that the solution exists globally or blows up in finite time with low initial energy and critical energy. Further a finite time blow up result of the solution with high initial energy is proved.

Ion-induced particle desorption in time-of-flight medium energy ion scattering

Abstract Secondary ions emitted from solids upon ion impact are studied in a time-of-flight medium energy ion scattering (ToF-MEIS) set-up. In order to investigate characteristics of the emission processes and to evaluate the potential for surface and thin film analysis, experiments employing TiN and Al samples were conducted. The ejected ions exhibit a low initial kinetic energy of a few eV, thus, requiring a sufficiently high acceleration voltage for detection. Molecular and atomic ions of different charge states originating both from surface contaminations and the sample material are found, and relative yields of several species were determined. Experimental evidence that points towards a predominantly electronic sputtering process is presented. For emitted Ti target atoms an additional nuclear sputtering component is suggested.

Fast-charging high-energy lithium-ion batteries via implantation of amorphous silicon nanolayer in edge-plane activated graphite anodes

As fast-charging lithium-ion batteries turn into increasingly important components in forthcoming applications, various strategies have been devoted to the development of high-rate anodes. However, despite vigorous efforts, the low initial Coulombic efficiency and poor volumetric energy density with insufficient electrode conditions remain critical challenges that have to be addressed. Herein, we demonstrate a hybrid anode via incorporation of a uniformly implanted amorphous silicon nanolayer and edge-site-activated graphite. This architecture succeeds in improving lithium ion transport and minimizing initial capacity losses even with increase in energy density. As a result, the hybrid anode exhibits an exceptional initial Coulombic efficiency (93.8%) and predominant fast-charging behavior with industrial electrode conditions. As a result, a full-cell demonstrates a higher energy density (≥1060 Wh l−1) without any trace of lithium plating at a harsh charging current density (10.2 mA cm−2) and 1.5 times faster charging than that of conventional graphite.

Fracture Toughness of Z3CN20.09M Cast Stainless Steel with Long-Term Thermal Aging

Accelerated thermal aging tests were performed at 400 °C for nearly 18,000 h on Z3CN20.09M cast stainless steel which was used for primary coolant pipes of nuclear power plants. A series of Charpy impact tests were conducted on Z3CN20.09M after different long-term thermal aging time. The test results indicated that the Charpy impact energy of Z3CN20.09M cast stainless steel decreased rapidly at an early stage and then almost saturated after thermal aging of 10,000 h. Furthermore, J-resistance curves were measured for CT specimens of longitudinal and circumferential pipe orientations. It showed that there was no obvious difference in the fracture characteristics of Z3CN20.09M in different sampling directions. In addition, the observed stretch zone width (SZW) revealed that the value of initiation fracture toughness J SZW was significantly lower than that of fracture toughness J IC, indicating a low actual crack initiation energy due to long-term thermal aging.

The ASLO Storytellers Series Connecting with Our Keiki (Hawaiian for “Children”)

The ASLO Storytellers Series Connecting with Our Keiki (Hawaiian for “Children”)

Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations

A linear oscillator simultaneously subjected to stochastic forcing and parametric excitation is considered. The time required for this system to evolve from a low initial energy level until a higher energy state for the first time is a random variable. Its expectation satisfies the Pontryagin equation of the problem, which is solved with the asymptotic expansion method developed by Khasminskii. This allowed deriving closed-form expressions for the expected first passage time. A comprehensive parameter analysis of these solutions is performed. Beside identifying the important dimensionless groups governing the problem, it also highlights three important regimes which are called incubation, multiplicative and additive because of their specific features. Those three regimes are discussed with the parameters of the problem.

Global non-existence for some nonlinear wave equations with damping and source terms in an inhomogeneous medium

In this paper we study the initial value problem for the nonlinear wave equation with damping and source terms \t\t\tutt−ρ(x)−1Δu+ut+m2u=f(u)\\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}$$ u_{tt}-\\rho(x)^{-1}\\varDelta u+u_{t}+m^{2}u=f(u) $$\\end{document} with some rho(x) and f(u) on the whole space mathbb{R}^{n} (ngeq 3).For the low initial energy case, which is the non-positive initial energy, based on a concavity argument we prove the blow-up result. As for the high initial energy case, we give sufficient conditions of the initial data such that the corresponding solution blows up in finite time. In other words, our results imply a complete blow-up theorem in the sense of the initial energy, -infty< E(0)<+infty.

Plastic analyses of thin-walled steel honeycombs with re-entrant deformation style

To get an ideal energy absorbing structure, which has low initial peak force but high energy absorbing efficiency, non-entrant and re-entrant honeycombs have re-entrant deformation styles were designed and fabricated. In-plane compression experiments and numerical simulations were carried out to reveal the progressive crushing process and the energy absorption level. In the crushing, the re-entrant honeycomb contracts laterally and has the potential to turn to a triangular lattice structure. The initial yield force of these honeycombs is not high but the stress after yield keeps high level and the plateau in stable. It is found to get a stable deformation curve the vertical wall must be thicker. Based on the crushing modes, plastic models were built to predict the mean crushing force (MCF) and they fit the test data very well. It is found that the re-entrant honeycomb has better energy absorption than the non-entrant honeycomb. Adopting re-entrant design, the MCF is even greater than the peak force, which benefits engineering design for energy absorbing structures in devices which requires smaller peak force but greater energy absorption.