Energy decay for a class of nonlinear wave equations with a critical potential type of damping

The Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│）-1 and a nonlinearity │u│p-1u is studied.The total energy decay estimates of the global solutions are obtained by using multiplier techniques to establish identity ddtE（t）＋F（t）=0 and skillfully selecting f（t）,g（t）,h（t）when the initial data have a compact support.Using the similar method,the Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│＋t）-1 and a nonlinearity │u│p-1u is studied,similar solutions are obtained when the initial data have a compact support.

MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved. Furthermore, by the improved inequalities and the linking theorem, in a new k-order SobolevHardy space, we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△（k）u：=-△u-（N-2）2/4u/｜x｜2-1/4k-1∑im1u/｜x｜2（ln（i）R/｜x｜2=f（x,u）,x∈Ω,u=0,x∈Ω,where 0∈ΩBa（0）RN,n≥3,ln）i）=6jm1ln（j）,and R=ae（k-1）,where e（0）=1,e（j）=ee（j=1）for j≥1,ln（1）=ln,ln（j）=lnln（j-1）for j≥2.Besides,positive and negative solutions are obtained by a variant mountain pass theorem.

Nonlinear Biharmonic Equations with Critical Potential

In this paper, we study two semilinear singular biharmonic equations： one with subcritical exponent and critical potential, another with sub-critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile, we have compared the changes of the critical dimensions in singular and non-singular cases, and we get an interesting result.

Singular Quasilinear Elliptic Problems with Indefinite Weights and Critical Potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN, is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone＇s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.

EXISTENCE RESULTS FOR DEGENERATE ELLIPTIC EQUATIONS WITH CRITICAL CONE SOBOLEV EXPONENTS

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.

Critical Length of Double-Walled Carbon Nanotubes Based Oscillators

The critical lengths of an oscillator based on double-walled carbon nanotubes(DWCNTs)are studied by energy minimization and molecular dynamics simulation.Van der Waals(vdW)potential energy in DWCNTs is shown to be changed periodically with the lattice matching of the inner and outer tubes by using atomistic models with energy minimization method.If the coincidence length between the inner and outer tubes is long enough,the restoring force cannot drive the DWCNT to slide over the vdW potential barrier to assure the DWCNT acts as an oscillator.The critical coincidence lengths of the oscillators are predicted by a very simple equation and then confirmed with energy minimization method for both the zigzag/zigzag system and the armchair/armchair system.The critical length of the armchair/armchair system is much larger than that of the zigzag/zigzag system.The vdW potential energy fluctuation of the armchair/armchair system is weaker than that of the zigzag/zigzag system.So it is easier to slide over the barrier for the armchair/armchair system.The critical lengths of zigzag/zigzag DWCNTbased oscillator are found increasing along with temperature,by molecular dynamics simulations.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

A Reevaluation of the Critical Membrane Potential of the Effective Refractory Period in Guinea Pig Ventricular Fibres

The traditional critical membrane potential (CMP), -55—-60mV, which corresponds to effective refractory period (ERP), was anew investigated in guinea pig ventricular muscle fibres. The electrical and contractile responses to the stimulus during repolarization of action potential (AP), particularly from+10 to -60 mV, were observed. One third of 35 tested cells displayed testing action potential (TAP) and local response at≥-54 mV when they were stimulated by testing pulses in 37℃ normal Tyrode's solution. Potential level of TAP which occurred earliest was at -30 mV and that of local response which appeared earliest was at 0 mV during repolarization among 95 systematic tests. Most of the TAPs belonged to the slow response potential type. The ratio of TAP evoked at ≥-54 mV initial membrane potential (IMP) was as high as 86% when the experiment was carried out in 37℃ 1.5 mmol KC1/L Tyrode's solution. In view of distribution of IMPs of TAPs, the CMP of ERP in guinea pig ventricular muscle fibres was more positive than traditional CMP measured by Hoffman et al. in dog, sheep Purkinje fibres and had a quite changeable range. The CMP of every cell in ventricular muscle was not all the same, and their CMPs approximated to normal distribution. There was no sharp line separating ERP from relative refractory period in myocardium. Higher temperature and low [K]_0 were the important factors elevating CMP of ERP.

Vibration and Buckling Analysis of Piezoelectric Nanowires Based on Surface Energy Density

In this paper,the influences of surface effects on free transverse vibration and buckling of piezoelectric nanowires are investigated by surface energy density elasticity theory.Analytical relations are given for the natural frequencies of nanowires by taking into account the effects of surface energy density and surface relaxation parameter.By implementing this theory with consideration of surface effects under clamped-clamped boundary conditions,the natural frequencies of nanowires are calculated.It is shown that the natural frequency depends on both the surface effects and piezoelectricity.A closed-form solution is also obtained to calculate the critical buckling voltage.This study is expected to provide useful insights for the design of piezoelectric nanowire-based nanodevices.