EXPONENTIAL DECAY FOR A NONLINEAR VISCOELASTIC EQUATION WITH SINGULAR KERNELS

The nonlinear viscoelastic wave equation ｜μt｜^pμtt-△μ-μutt＋∫^t0g（t-s）△μ（s）ds＋｜μ｜^pU=0,in a bounded domain with initial conditions and Dirichlet boundary conditions is consid- ered. We prove that, for a class of kernels 9 which is singular at zero, the exponential decay rate of the solution energy. The result is obtained by introducing an appropriate Lyapounov functional and using energy method similar to the work of Tatar in 2009. This work improves earlier results.

Exponential decay law of acoustic emission and microseismic activities caused by disturbances associated with multilevel loading and mining blast

To investigate decay law of acoustic emission and microseismic activities caused by disturbances associated with multilevel loading and mining blast,a new exponential decay(ED)law was proposed.The results show that the micro-fracture activity decay law after multistage stress loading and blasting disturbance conforms to the ED model,in which the sum of A and n represents the number of initial micro-fracture events,and n represents the level of background micro-fracture events.The ED model can describe the number of initial micro-fracture events with a deviation less than 10%.The ED model outperforms the traditional aftershock models in describing the micro-fracture event decay law in the three cases considered.The ED model can provide a reference for selecting the blasting interval in mines,which is of great significance to maintain the stability of the surrounding rock and ensure safe production in metal mines.

GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM

We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small.Note that although the system degenerates near vacuum,there is no need to require compatibility conditions for the initial data via time-weighted techniques.

GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0 〈 E（0） 〈 d. At last we show that the energy will grow up as an exponential function as time goes to infinity, provided the initial data is large enough or E（0） 〈 0.

Global Existence and Exponential Decay for a Nonlinear Viscoelastic Equation with Nonlinear Damping

In this paper we investigate a nonlinear viscoelastic equation with nonlinear damping. Global existence of weak solutions and uniform decay of the energy have been established. The Faedo-Galerkin method and the perturbed energy method are employed to obtain the results.

Exponential decay of expansive constants

A map f on a compact metric space is expansive if and only if fn is expansive.We study the exponential rate of decay of the expansive constant of fn and find some of its relations with other quantities about the dynamics,such as box dimension and topological entropy.

Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered. Analyticity in time for a class of solutions taking values in a Gevrey class of functions is proven. Exponential decay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by the exponential decay are also obtained.

Short-term forecast in the early stage of the COVID-19 outbreak in Italy. Application of a weighted and cumulative average daily growth rate to an exponential decay model

To estimate the size of the novel coronavirus(COVID-19)outbreak in the early stage in Italy,this paper introduces the cumulated and weighted average daily growth rate(WR)to evaluate an epidemic curve.On the basis of an exponential decay model(EDM),we provide estimations of the WR in four-time intervals from February 27 to April 07,2020.By calibrating the parameters of the EDM to the reported data in Hubei Province of China,we also attempt to forecast the evolution of the outbreak.We compare the EDM applied to WR and the Gompertz model,which is based on exponential decay and is often used to estimate cumulative events.Specifically,we assess the performance of each model to short-term forecast of the epidemic,and to predict the final epidemic size.Based on the official counts for confirmed cases,the model applied to data from February 27 until the 17th of March estimate that the cumulative number of infected in Italy could reach 131,280(with a credibility interval 71,415-263,501)by April 25(credibility interval April 12 to May 3).With the data available until the 24st of March the peak date should be reached on May 3(April 23 to May 23)with 197,179 cumulative infections expected(130,033e315,269);with data available until the 31st of March the peak should be reached on May 4(April 25 to May 18)with 202,210 cumulative infections expected(155.235 e270,737);with data available until the 07st of April the peak should be reached on May 3(April 26 toMay 11)with 191,586(160,861-232,023)cumulative infections expected.Based on the average mean absolute percentage error(MAPE),cumulated infections forecasts provided by the EDM applied to WR performed better across all scenarios than the Gompertz model.An exponential decay model applied to the cumulated and weighted average daily growth rate appears to be useful in estimating the number of cases and peak of the COVID-19 outbreak in Italy and the model was more reliable in the exponential growth phase.

Exponential Decay in Time of Density of One-dimensional Quantum Navier-Stokes Equations

The long-time behavior of the particle density of the compressible quantum Navier-Stokes equations in one space dimension is studied. It is shown that the particle density converges exponentially fast to the constant thermal equilibrium state as the time tends to infinity, the decay rate is also obtained. The results hold regardless of either the bigger of the scaled Planck constant or the viscosity constant. This improves the decay results of [5] by removing the crucial assumption that the scaled Planck constant is bigger than the viscosity constant. The proof is based on the entropy dissipation method and the Bresch-Desjardins type of entropy.

Remark on Exponential Decay of Ground States for N-Laplacian Equations

We study exponential decay property of radial ground states to a class of N-Laplacian elliptic equations in the whole space R^N. Their decay rates as /x/→∞ are obtained explicitly.

EXPONENTIAL DECAY DOMAIN OF ENERGY FOR WAVE EQUATION UNDER FEEDBACK CONTROL

This paper deals with the energy estimate of wave equation with boundary and distributedfeedback action. It is shown that the energy of the system decays exponentially if the feedback parameters are in some domain (i.e., the so-called exponential decay domain). And thedependence on the feedback parameters of the energy exponeatial decay estimate is obtained.

Potential Response Analysis of Exponentially Decaying Polarizing Current of Electrode and its Use in the Study of Stress Corrosion Cracking(SCC)of Stainless Steel

A study on the potential response of exponentially decaying polarizing current of electrode was car- ried out.The appearance of critical point of the potential-time response of exponentially decaying current was ensured from theoretical analysis,and this is the theoretical foundation of the critical point method for the measurement of corrosion current of metals.The comparison of the corrosion currents measured by the critical point and static methods for the system of 321 stainless steel in 0.5N HCI+0.5N NaCI solution at static state shows that the results agree very well.Finally.the tran- sient corrosion currents of 321 stainless steel in 0.5N HCI+0.5N NaCl solution at different strain level are listed.

Efficiency of a Modular Cleanroom for Space Applications

A prototype cleanroom for hazardous testing and handling of satellites prior to launcher encapsulation,satisfying the ISO8 standard has been designed and analyzed in terms of performances.Unsteady Reynolds Averaged Navier-Stokes(URANS)models have been used to study the related flow field and particulate matter(PM)dispersion.The outcomes of the URANS models have been validated through comparison with equivalent large-eddy simulations.Special attention has been paid to the location and shape of the air intakes and their orientation in space,in order to balance the PM convection and diffusion inside the cleanroom.Forming a cyclone-type flow pattern inside the cleanroom is a key to maintaining a high ventilation efficiency.

General decay of energy to a nonlinear viscoelastic two-dimensional beam

A viscoelastic beam in a two-dimensional space is considered with nonlinear tension. A boundary feedback is applied at the right boundary of the beam to suppress the undesirable vibration. The well-posedness of the problem is established. With the multiplier method, a uniform decay result is proven.

ASYMPTOTIC BEHAVIOUR AND EXPONENTIAL STABILITY FOR THERMOELASTIC PROBLEM WITH LOCALIZED DAMPING

A semi-linear thermoelastic problem with localized damping is considered, which is one of the most important mathematical models in material science. The existence and decays exponentially to zero of solution of this problem are obtained. Moreover, the existence of absorbing sets is achieved in the non-homogeneous case. The result indicates that the system which we studied here is asymptotic stability.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Calibration of Soil Electromagnetic Conductivity in Inverted Salinity Profiles with an Integration Method

Various calibration methods have been propounded to determine profiles of apparent bulk soil electrical conductivity (ECa) and soil electrical conductivity of a saturated soil paste extract (ECe) or a 1:5 soil water extract (EC1:5) using an electromagnetic induction instrument (EM38). The modeled coefficients, one of the successful and classical methods hitherto, were chosen to calibrate the EM38 measurements of the inverted salinity profiles of characteristic coastal saline soils at selected sites of Xincao Farm, Jiangsu Province, China. However, this method required three parameters for each depth layer. An integration approach, based on an exponential decay profile model, was proposed and the model was fitted to all the calibration sites. The obtained model can then be used to predict EC1:5 at a certain depth from electromagnetic measurements made using the EM38 device positioned in horizontal and vertical positions at the soil surface. This exponential decay model predicted the EC1:5 well according to the results of a one-way analysis of variance, and the further comparison indicated that the modeled coefficients appeared to be slightly superior to, but not statistically different from, this exponential decay model. Nevertheless, this exponential decay model was more significant and practical because it depended on less empirical parameters and could be used to perform point predictions of EC1:5 continuously with depth.

Development of 3D top coal caving angle model for fully mechanized extra-thick coal seam mining

During high-intensity,fully mechanized mining of extra-thick coal seam,the top coal would cave to a certain 3D form.Based on the data collected during drilling,a 3D model of top coal caving surface space was established to determine the relationship between the location of the stope roof and the caving surface,enabling the mathematical computation of the top caving angle(φ).The drilling method was employed to measure the top caving angle on two extra-thick fully mechanized coal caving faces under the conditions of three geological structures,namely,no geological structure,igneous rock structure,and fault structure.The results show that the value of top caving angle could be accurately estimated on-site with the 9-parameter 3D top coal caving surface model built with the drilling method.This method is a novel on-site measurement that can be easily applied.Our findings reveal that the characteristics of the coal-rock in the two mining faces are different;yet their caving angles follow the ruleφ_(igneous rock structure)<φ_(no geological structure)<φ_(fault structure).Finally,through the data fitting with two indexes(the top coal uniaxial compressive strength and the top caving angle),it is found that the relationship between the two indexes satisfies an exponential decay function.

LARGE-TIME BEHAVIOR OF SOLUTIONS OF QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

A one-dimensional quantum hydrodynamic model （or quantum Euler-Poisson system） for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of t he steadystate solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.

STABILIZATION OF VIBRATING BEAM BY VELOCITY FEEDBACK CONTROL

A flexible structure consisting of a Euler-Bernoulli beam with co-located sensors and actuators is considered. The control is a shear force in proportion to velocity. It is known that uniform exponential stability can be achieved with velocity feedback. A sensitivity asymptotic analysis of the system's eigenvalues and eigenfunctions is set up. The authors prove that, for K-1 epsilon (0, + infinity), all of the generalized eigenvectors of A form a Riesz basis of H. It is also proved that the optimal exponential decay rate can be obtained from the spectrum of the system for 0 < K-1 < + infinity.