Nonsingle-valley Continuous Solutions of p-order Feigenbaum's Functional Equation

The author considers the Feigenbaum's functional equation fp(λx) = λf(x) for each p > 2. The existence of nonsingle-valley continuous solutions to this equation is discussed and a feasible method to construct such solutions is given.

PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

Results on the existence of piecewise continuous solutions for two classes of initial value problems of impulsive singular fractional differential equations are obtained.

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞）, there exist a weak solution （v, θ） of the 3d Boussinesq equations such that （v, 8） ∈ Cβ（T3 × [0, T]） with e（t） = ∫T3 ｜v（x, t）｜2dx for all t ∈ [0, T]. This extend the result of [2] about Onsager＇s conjecture into Boussinesq equation and improve our previous result in [30].

EXISTENCE AND NONEXISTENCE OF GLOBAL CONTINUOUS SOLUTIONS TO RIEMANN PROBLEM FOR DAMPED P-SYSTEM

In this paper, we extend the result in [16] to general p(v). We prove that, under condition (M), when P greater-than-or-equal-to 3/2, where P=pp triple overdot/p2, there exists a unique global continuous solution to the Riemann problem (E), (R), whose structure is similar to the local solution. When 1 < P* less-than-or-equal-to P* < 5/4, or P* = P* = 5/4, or 5 < P* less-than-or-equal-to P* < 3/2 where P*-inf/v P and P* = sup/v P for all v under consideration, if at least one of the initial centered rarefaction waves is sufficiently strong, then the solution must be breakdown in a finite time.

Lipschitz Continuous Solutions to the Cauchy Problem for Quasi-linear Hyperbolic Systems

Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.

PERIODIC SOLUTIONS OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

The sufficient condition for the existence of2 π- periodic solutions of the following third- order functional differential equations with variable coefficients a(t) x (t) +bx″2 k- 1(t) +cx′2 k- 1(t) + 2 k- 1 i=1 cixi(t) +g(x(t-τ) ) =p(t) =p(t+2π) is obtained.The approach is based on the abstract continuation theorem from Mawhin and the a- priori estimate of periodic solutions

THE CONTINUAL DIFFERENTIABLE PEAK-UNIMODAL SOLUTIONS OF FEIGENBAUM’S FUNCTIONAL EQUATIONS

For the famous Feigenbaum's equations, in this paper, we established its constructive theorem of the peak-unimodal, then we found out other paths to explore the peak-unimodal solutions. For example, we proceed on the direction to try the non-symmetrical continuous peak-unimodal solutions and C1 solutions.

THE CONVERGENT RATE OF VISCOSITY METHOD FOR A VARIANT NON-ISENTROPIC SYSTEM OF POLYTROPIC GAS

In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obtain a global Holder continuous solution and the convergent rate of the viscosity method for the Cauchy problem of the variant nonisentropic system of polytropic gas for any adiabatic exponentγ>1.

Holder continuous weak solutions of the 2D Boussinesq equation with thermal diffusion

We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].

Corrosion behavior of Ti60 alloy under continuous NaCl solution spraying at 600℃

The corrosion behavior of Ti60 alloy was investigated under continuous NaCl solution spraying at 600℃.Results indicate that the corrosion rate of the Ti60 alloy is lower than that obtained with the solid NaCl deposit film in HO+O.The outer corrosion products are compact layers mainly containing NaTiOand nanocrystalline TiO.The inner layer is mainly composed of TiO,Ti O and SnO.TiO has a lamellar structure,and TiO has a strip-shaped structure.A mixed corrosion mechanism including classic oxidation and active oxidation occurred under NaCl solution spraying environment:the first stage was controlled by the classic oxidation mechanism with the promotion by the active oxidation following parabolic curves,and the second stage was controlled by the active oxidation mechanism following a linear rise in the mass gain curve.The mixed corrosion mechanism between classic oxidation and active oxidation produced a relatively dense Ti oxides scale,which decreased the corrosion rate.

Characteristics of Solute Transport Continuously Released from Coastal Unconfined Aquifers under the Tidal Action Based on Laboratory Experiment

Most studies on solute transport in coastal aquifers affected by tides focus on the transport of instantaneous released solute,and there are few studies on continuously released solute affected by tides.In this study,the image monitoring method is used to establish the quantitative relationship between the concentration of the colored tracer and the hue value of the image,and the digital image is used to determine the tracer concentration distribution.Using image monitoring method laboratory experiments,quantitative analysis of the characteristics of continuously released solute transport in coastal unconfined aquifers under the tidal influence.Experiments show that the high tide inhibits the increase in the concentration of each point in the aquifer.Under the influence of tides,the solute plume retreats towards the land.During the low tide period,the solute plume migrates toward the sea again.And the solute plume will maintain a relatively stable shape after entering the aquifer for a long enough time.Ignoring the tidal effect seems to have little effect on the estimation of the position of the solute plume,but ignoring the tidal effect has a certain influence on the estimation of the dispersion range of the solute plume.No matter whether considering the tidal action,the final dispersion range of the solute plume is almost the same.But before the solute plume reaches a stable state,ignoring the tidal effect will lead to a smaller dispersion range of the solute plume.

Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ＇（.） is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].

Hlder Continuity of the Gradient of Solutions of Nonlinear Degenerate Parabolic Systems

§1. Introduction In this paper we consider the parabolic system (?)t/(?)ui-div(|▽u|p-2▽ui)=0(1≤i≤m), with p>1, where ui=ui(x, t), ▽=gradxand x varies in an open domain Ω(?)RN. The system is degenerate if p>2 or singular if 1<p<2. A vector function u=(u1, u2, …, um) defined in ΩT=Ω×[0, T] is called a solution of the system (1.1) if

The polynomial-like iterative equation for PM functions

The polynomial-like iterative equation is an important form of functional equations, in which iterates of the unknown function are linked in a linear combination. Most of known results were given for the given function to be monotone. We discuss this equation for continuous solutions in the case that the given function is a PM(piecewise monotone) function, a special class of non-monotonic functions. Using extension method, we give a general construction of solutions for the polynomial-like iterative equation.

ON THE FEIGENBAUM'S FUNCTIONAL EQUATION_f^P (λx )=λf (x)

The author considers the Feigenbaum's functional equation fP(λx) =λf(x) for each p ≥ 2.The existence of even unimodal C1 solutions to this equation is discussed and a feasible methodto construct such solutions is given.

ON SOME DISCRETE INEQUALITIES USEFUL IN THE THEORY OF CERTAINPARTIAL FINITE DIFFERENCE EQUATIONS

The aim of this paper is to establish some new discrete inequalities in two independent variables which can be used as handy tools.in the theory of certain fourth order partial finite difference equations. The analysis used in the proof is elementary and the results established provide new estimates for these types of inequalities.AMS (MOS) Subject Classification (1991 ): Primary 26D15.

Isothermal section of Gd-Sm-Co ternary system at 773K

The isothermal section of the phase diagram of the Gd-Sm-Co ternary system at 773 K was investigated by X-ray powder diffraction （XRD）, differential thermal analysis （DTA）, optical microscopy and scanning electron microscopy （SEM） techniques. The result shows that the isothermal section consists of 12 single-phase regions, 16 two-phase regions and 5 three-phase regions. Five pairs of corresponding compounds of Gd-Co and Sin-Co systems, i.e., Gd2Co17 and Sm2Co^7, Gd2Co7 and Sm2Co7, GdCo3 and SmC03, GdCo： and SmCo：, Gd3Co and Sm3Co form continuous series of solid solutions. The maximum solid solubility of Sm in Gd4Co3 arid Gdl2Co7 were about 7.2 at.% and 47.8 at.% Sm, respectively. The maximum solid solubility of Gd in Sm5Col9 and SmsCo2 were about 4.7 at.% and 7.6 at.% Gd, respectively. The binary compounds Sm9Co4, GdCos and SmCo5 were not observed at 773 K. No ternary compound was found.