Global Bounded Solutions for the Keller-Segel Chemotaxis System with Singular Sensitivity

In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity (0.1) is considered in a bounded domain with smooth boundary, Ω ⊂Rn (n ≥ 1), where d1 > 0, d2 > 0 with parameter χ ∈ R. When d1 = d2 + χ, satisfying for all initial data 0 ≤ n0 ∈ C0 and 0 < v0∈ W1,∞ (Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω × (0, ∞).

THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE

In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.

BIFURCATION OF BOUNDED SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER

Using the notion of an isolated invariant set and an isolating block, an existence criterion of bifurcation points of nonstationary bounded solutions to ordinary differential systems depending on a parameter is given.

Continuous Dependence of Bounded Φ-variation Solutions on Parameter for a Class of Discontinuous Systems

The functions of bounded φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems. In this paper, by using Henstock-Kurzweil integral, we establish theorems of continuous dependence of bounded D-variation solutions on parameter for a class of discontinuous systems on the base of D-function. These results are essential generalizations of continuous dependence of bounded variation solutions on parameter for the systems.

BOUNDED SOLUTIONS AND PERIODIC SOLUTIONS OF VISCOUS POLYTROPIC GAS EQUATIONS

A piston problem of viscous polytropit gas equations is discussed.It is shown that the global solution is bounded uniformly in time if the piston motion is bounded and that if the piston motion is periodic in time,then there exists a periodic solution to the piston problem with the same period.

Continuous Dependence of Bounded Φ-Variation Solutions on Parameters for Kurzweil Equations

The continuous dependence of bounded Φ-variation solutions on parameters for Kurzweil equations are established by using the functions of bounded Φ- variation that were introduced by Musielak-Orlice. These results are essential generalizations of continuous dependence of bounded variation solutions on parameters for Kurzweil equations.

BOUNDED SOLUTIONS AND PERIODIC SOLUTIONS OF VISCOUS POLYTROPIC GAS EQUATIONS

A piston problem of viscous polytropic gas equations is discussed. It is shown that the global solution is bounded uniformly in time if the piston motion is bounded and that if the piston motion is periodic in time, then there exists a periodic solution to the piston problem with the same period.

Boundedness of Solutions for Elliptic Variational Inequalities

The boundedness is proved under more general structural conditions to solutions of elliptic variational inequalities and a priori estimates are obtained to maximum modulus of solutions for some special cases.

BOUNDED TRAVELING WAVE SOLUTIONS OF VARIANT BOUSSINESQ EQUATION WITH A DISSIPATION TERM AND DISSIPATION EFFECT

This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^＊ which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if ｜r｜≥ r^＊; while they appear as damped oscillatory waves if ｜r｜ 〈 r^＊. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u（ξ） in the global phase portraits, and then obtain the approximate solutions （u（ξ）, H（ξ））. Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.

NUMBER OF BOUNDED SOLUTIONS IN CRITICAL CASE

In this paper, using the method of linearization, we determine the number of bounded solutions of ordinary differential equations in critical case.

Subspaces for weak mild solutions of the second order abstract differential equation

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.

THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.

SIGN-CHANGING SOLUTIONS FOR SCHRDINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u ＋ λV（x）u = K（x）｜u｜^p-2u x∈R^N, u→0 as ｜x｜→ ＋∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V（x） and K（x） satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].

Upper Bound Solutions for Uplift Ultimate Bearing Capacity of Suction Caisson Foundation

Suction caisson foundation derives most of their uplift resistance from passive suction developed during the pullout movement. It was observed that the passive suction generated in soil at the bottom of the caisson and the failure mode of suction caisson foundation subjecting pullout loading behaves as a reverse compression failure mechanism.The upper bound theorems have been proved to be a powerful method to find the critical failure mechanism and critical load associated with foundations, buried caissons and other geotechnical structures. However, limited attempts have been reported to estimate the uplift bearing capacity of the suction caisson foundation using the upper bound solution. In this paper, both reverse failure mechanisms from Prandtl and Hill were adopted as the failure mechanisms for the computation of the uplift bearing capacity of the suction caisson. New equations were proposed based on both failure mechanisms to estimate the pullout capacity of the suction caisson. The proposed equations were verified by the test results and experimental data from published literature. And the two solutions agree reasonably well with the other test results. It can be proved that both failure mechanisms are reasonably and more consistent with the actual force condition.

Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation X^s+ A~*X^(-t) A = Q

The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.

POSITIVE ENTIRE SOLUTIONS TO SINGULAR SEMILINEAR ELLIPTIC EQUATIONS OF MIXED TYPE

In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.

Eventually vanished solutions of a forced Liénard system

In this paper, we aim to find eventually vanished solutions, a special class of bounded solutions which tend to 0 as t → ±∞）, to a Lienard system with a time-dependent force. Since it is not a Hamiltonian system with small perturbations, the well-known Melnikov method is not applicable to the determination of the existence of eventually vanished solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. Difficulties caused by the non- Hamiltonian form are overcome by applying the Schauder＇s fixed point theorem. We show that the sequence of the periodic solutions has an accumulation giving an eventually vanished solution of the forced Lienard system.

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

REMARKS ON BOUNDS ON THE DISCREPANCY OF APPROXIMATE SOLUTIONS CONSTRUCTED BY GODUNOV'S SCHEME

The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.

Recrystallization of freezable bound water in aqueous solutions of medium concentrations

For aqueous solutions with freezable bound water, vitrification and recrystallization are mingled, which brings difficulty to application and misleads the interpretation of relevant experiments. Here, we report a quantification scheme for the freezable bound water based on the water-content dependence of glass transition temperature, by which also the concentration range for the solutions that may undergo recrystallization finds a clear definition. Furthermore, we find that depending on the amount of the freezable bound water, different temperature protocols should be devised to achieve a complete recrystallization. Our results may be helpful for understanding the dynamics of supercooled aqueous solutions and for improving their manipulation in various industries.