THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

Unveiling the brain’s symphony:exploring the necessity and sufficiency of neural networks in behavior control

Since the pioneering work by Broca and Wernicke in the 19th century,who examined individuals with brain lesions to associate them with specific behaviors,it was evident that behaviors are complex and cannot be fully attributable to specific brain areas alone.Instead,they involve connectivity among brain areas,whether close or distant.At that time,this approach was considered the optimal way to dissect brain circuitry and function.These pioneering efforts opened the field to explore the necessity or sufficiency of brain areas in controlling behavior and hence dissecting brain function.However,the connectivity of the brain and the mechanisms through which various brain regions regulate specific behaviors,either individually or collaboratively,remain largely elusive.Utilizing animal models,researchers have endeavored to unravel the necessity or sufficiency of specific brain areas in influencing behavior;however,no clear associations have been firmly established.

Increased excitatory amino acid transporter 2 levels in basolateral amygdala astrocytes mediate chronic stress–induced anxiety-like behavior

The conventional perception of astrocytes as mere supportive cells within the brain has recently been called into question by empirical evidence, which has revealed their active involvement in regulating brain function and encoding behaviors associated with emotions.Specifically, astrocytes in the basolateral amygdala have been found to play a role in the modulation of anxiety-like behaviors triggered by chronic stress. Nevertheless, the precise molecular mechanisms by which basolateral amygdala astrocytes regulate chronic stress–induced anxiety-like behaviors remain to be fully elucidated. In this study, we found that in a mouse model of anxiety triggered by unpredictable chronic mild stress, the expression of excitatory amino acid transporter 2 was upregulated in the basolateral amygdala. Interestingly, our findings indicate that the targeted knockdown of excitatory amino acid transporter 2 specifically within the basolateral amygdala astrocytes was able to rescue the anxiety-like behavior in mice subjected to stress. Furthermore, we found that the overexpression of excitatory amino acid transporter 2 in the basolateral amygdala, whether achieved through intracranial administration of excitatory amino acid transporter 2agonists or through injection of excitatory amino acid transporter 2-overexpressing viruses with GfaABC1D promoters, evoked anxiety-like behavior in mice. Our single-nucleus RNA sequencing analysis further confirmed that chronic stress induced an upregulation of excitatory amino acid transporter 2 specifically in astrocytes in the basolateral amygdala. Moreover, through in vivo calcium signal recordings, we found that the frequency of calcium activity in the basolateral amygdala of mice subjected to chronic stress was higher compared with normal mice.After knocking down the expression of excitatory amino acid transporter 2 in the basolateral amygdala, the frequency of calcium activity was not significantly increased, and anxiety-like behavior was obviously mitigated. Additionally, administration of an excitatory amino acid transporter 2 inhibitor in the basolateral amygdala yielded a notable reduction in anxiety level among mice subjected to stress. These results suggest that basolateral amygdala astrocytic excitatory amino acid transporter 2 plays a role in in the regulation of unpredictable chronic mild stress-induced anxiety-like behavior by impacting the activity of local glutamatergic neurons, and targeting excitatory amino acid transporter 2 in the basolateral amygdala holds therapeutic promise for addressing anxiety disorders.

Asymptotic Behavior and Nonexistence of Positive Solutions of Delay Difference Equations *L

Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL REACTION-DIFFUSION SYSTEM

This paper deals with reaction-diffusion system with nonlocal source. It is proved that there exists a unique classical solution and the solution either exists globally or blows up in finite time. Furthermore, its blow-up set and asymptotic behavior are obtained provided that the solution blows up in finite time.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

COMPLEXITY OF ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE POROUS MEDIUM EQUATION WITH ABSORPTION

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut - △um ＋ yup = 0,where γ≥0,m〉 1and P〉m＋2/N We will show that if γ=0 and 0〈μ〈 2N/n（m-1）＋2 or γ 〉 0 and 1/p-1 〈 μ 〈 2N/N（m-1）＋2 then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S（RN）, there exists an initial-value u0 ∈ C（RN） with limx→∞ uo（x）= 0 such that φ is an w-limit point of the rescaled solutions tμ/2u（tβ, t）, Where β = 2-μ（m-1）/4.

ASYMPTOTIC BEHAVIOR FOR A CLASS OF ELLIPTICEQUIVALUED SURFACE BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS INTERFACE CONDITIONS

Spontaneous potential well-logging is one of the important techniques in petroleum exploitation. A spontaneous potential satisfies an elliptic equivalued surface boundary value problem with discontinuous interface conditions. In practice, the measuring electrode is so small that we can simplify the corresponding equivalued surface to a point. In this paper, we give a positive answer to this approximation process:when the equivalued surface shrinks to a point, the solution of the original equivalued surface boundary value problem converges to the solution of the corresponding limit boundary value problem.

On the Asymptotic Behavior of Certain Second Order Differential Equations

Two criteria for asymptotic behavior of certain second order differential equations are obtained. The results extend and improve Feng's theorem.

ASYMPTOTIC BEHAVIOR OF LINEAR ADVANCED DIFFERENTIAL EQUATIONS

In this article, we consider a general class of linear advanced differential equa- tions, and obtain explicitly sufficient conditions of convergence and exponential convergence to zero. A necessary condition is provided as well.

Asymptotic Behavior of a Stochastic SIRS Model with Non-linear Incidence and Levy Jumps

A stochastic susceptible-infective-recovered-susceptible( SIRS) model with non-linear incidence and Levy jumps was considered. Under certain conditions, the SIRS had a global positive solution. The stochastically ultimate boundedness of the solution of the model was obtained by using the method of Lyapunov function and the generalized Ito's formula. At last,asymptotic behaviors of the solution were discussed according to the value of R0. If R0< 1,the solution of the model oscillates around a steady state, which is the diseases free equilibrium of the corresponding deterministic model,and if R0> 1,it fluctuates around the endemic equilibrium of the deterministic model.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.

ASYMPTOTIC BEHAVIOR FOR COMMUTATIVE SEMIGROUPS OF ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS

This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak convergence theorems for almostorbits of this class of commutative semigroups. The author improves, extends and develops some recent and earlier results.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

In this paper,we study the initial-boundary value problem for a class of singular parabolic equations.Under some conditions,we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method.As a byproduct,we prove the existence of solutions to some problems with gradient terms,which blow up on the boundary.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u =φ（r）u^p-1, u 〉 0 in R^N, u ∈ D^1,2（R^N）, where N ≥ 3, x = （x^1,z） ∈ R^K×R^N-K,2 ≤ K ≤ N,r = ｜x′｜. It is proved that for 2（N-s）/（N-2） 〈 p 〈 2^＊ = 2N/（N - 2）, 0 〈 s 〈 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p = 2^＊, the above equation does not have a ground state solution but a cylindrically symmetric.solution, and when p close to 2^＊, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2＊, the ground state solution Up has a unique maximum point xp = （x′p, Zp） and as p → 2^＊, ｜x′p｜ → r0 which attains the maximum of φ on R^N. The asymptotic behavior of ground state solution Up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2^＊.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR AN 1-D COMPRESSIBLE ENERGY TRANSPORT MODEL

In this article, the global existence and the large time behavior of smooth solutions to the initial boundary value problem for a degenerate compressible energy transport model are established.

ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCAR DISK INTO BOUNDED SYMMETRIC DOMAINS

In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N＋1, and that the irreducible component of V ∩ （△ × Ω） containing V0 is the graph of a proper holomorphic isometric embedding F ： A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.

Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

We study a nonlinear equation in the half-space with a Hardy potential,specifically,−Δ_(p)u=λu^(p−1)x_(1)^(p)−x_(1)^(θ)f(u)in T,where Δp stands for the p-Laplacian operator defined by Δ_(p)u=div(∣Δu∣^(p−2)Δu),p>1,θ>−p,and T is a half-space{x_(1)>0}.When λ>Θ(where Θ is the Hardy constant),we show that under suitable conditions on f andθ,the equation has a unique positive solution.Moreover,the exact behavior of the unique positive solution as x_(1)→0^(+),and the symmetric property of the positive solution are obtained.

Asymptotic Behavior of Solutions for a Periodic Transport Equation with Delay

This paper analyzes a class of first order partial differential equations with delay (a model for the blood production system). The asymptotic behavior of solutions are studied.