THE EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR EVOLUTION EQUATIONS AND APPLICATIONS TO P. D. E.

The main purpose of this paper is to discuss the existence and asymptotic behavior of solutions for [GRAPHICS] and for which the sufficient conditions of asymptotic behavior are obtained and the restriction for the existence is reduced.

Asymptotic Behavior of Solutions of Retarded Differential Equations

A sufficient condition is obtained for every solution of the nonlinear retarded differential equationx'(t) +f(t,x(t-τ)) =0to tend to zero as t→∞ , which extends and improves the corresponding results obtained by Ladas, Sficas and Gopalsamy.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A CLASS OF HIGHER ORDER DIFFERENCE EQUATIONS

In this paper,we study the asymptotic behavior of solutions to a class of higher order difference equations.With the aid of the discrete inequality,we obtain some sufficient conditions which ensure that all the solutions to the equation are some high order of infinities,and also that some conditions which guarantee that every oscillatory solution to the equation has the property that the i order L operator of it tends to infinity when its independent variable tends to zero.

Asymptotic behavior of solutions for some chemical reaction models

IN the process of forecasts, analyses and numerical treatments of the ground water resource, we often meet with various chemical reaction models. One type of them is three kinds of chemical substance M1, M2 and M3 which can react with each other to produce two new kinds of other chemical compounds: （M2）n （M1）m and （M3）r （M1）κ at the same time. Usually, these reactions are irreversible and they have the following forms:

ASYMPTOTIC BEHAVIOR OF HIGHER ORDER NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

In this paper, we study a class of higher order nonlinear integro-diferential equations with deviating arguments. With the aid of the integral inequality, we obtain some sufcient conditions under which all solutions to the equation have some asymptotic behavior.

ASYMPTOTIC BEHAVIOR OF A CLASS OF NONLINEAR DIFFERENCE EQUATIONS OF nTH ORDER

In this paper, we study the asymptotic behavior of the nth order nonlinear differenceequationwhere △ is the forward differenceoperator, i.e.,

ENERGY DECAY OF SOLUTIONS TO A NONLINEAR EQUATION ARISING FROM ELASTIC WAVEGUIDE MODEL

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for a nonlinear equation arising from an elastic waveguide model. We prove that under rather mild conditions the initial boundary value problem possesses global solutions which decay at an exponential rate.

ATTRACTIVITY PROPERTIES OF SOLUTIONS FOR DEGENERATE QUASILINEAR PARABOLIC EQUATIONS OF HIGHER ORDER

In this paper, we show some attractivity properties of the solutions (as t→∞)to degenerate quasilinear parabolic equations of higher order.

Existence and asymptotics of normalized solutions for the logarithmic Schrödinger system

This paper is concerned with the following logarithmic Schrodinger system:{-Δu_(1)+ω_(1)u_(1)=u_(1)u_(1)logu_(1)^(2)+2p/p+q|u_(2)|^(q)|u_(1)|^(p-2)u_(1),-Δu_(2)+ω_(2)u_(2)=u_(2)u_(2)log u_(2)^(2)+2q/p+q|u_(1)|^(p)|u_(2)|^(q-2)u_(2),∫_(Ω)|u_(i)|^(2)dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2)),where Ω=R^(N)or Ω■R^(N)(N≥3)is a bounded smooth domain,andω_(i)R,μ_(i),ρ_(i)>0 for i=1,2.Moreover,p,q≥1,and 2≤p+q≤2^(*),where 2^(*):=2N/N-2.By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u^(2),firstly,we provide a unified proof of the existence of the normalized ground state solution for all 2≤p+q≤2^(*).Secondly,we consider the stability of normalized ground state solutions.Finally,we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p+q approaches 2^(*).Notably,the uncertainty of the sign of u log u^(2)in(0,+∞)is one of the difficulties of this paper,and also one of the motivations we are interested in.In particular,we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations(i.e.,p+q=2^(*)).In addition,our study includes proving the existence of solutions to the logarithmic type Bréis-Nirenberg problem with and without the L^(2)-mass.constraint ∫_(Ω)|u_(i)|^(2)dx=ρ_(i)(i=1,2)by two different methods,respectively.Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrodinger system with logarithmic perturbations.

Global Helically Symmetric Solutions to 3D MHD Equations

We prove the existence and uniqueness of global strong solutions to the Cauchy problem of the three-dimensional magnetohydrodynamic equations in R3 when initial data are helically symmetric. Moreover, the large-time behavior of the strong solutions is obtained simultaneously.

Spatiotemporal patterns induced by cross-diffusion in predator prey model with prey herd shape effect

In this paper,we investigate a predator-prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions.First,the temporal behavior of the model has been investigated,where Hopf bifurcation has been obtained.Then,by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation,Turing instability,even Turing-Hopf bifurcation.Further,the impact of the prey herd shape on the spatiotemporal patterns has been discussed.Furthermore,by computing and analyzing the normal form associated with the Turing-Hopf bifurcation point,the spatiotemporal dynamics near the Turing-Hopf bifurcation point has been discussed and allso justified by some numerical simulations.