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.

THE REGULARITY CRITERIA OF WEAK SOLUTIONS TO 3D AXISYMMETRIC INCOMPRESSIBLE BOUSSINESQ EQUATIONS

In this paper,we obtain new regularity criteria for the weak solutions to the three dimensional axisymmetric incompressible Boussinesq equations.To be more precise,under some conditions on the swirling component of vorticity,we can conclude that the weak solutions are regular.

ON THE REGULARITY CRITERIA OF THE 3D NAVIER-STOKES EQUATIONS IN CRITICAL SPACES

Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, θ1u1, θ2u2, of velocity fields.

REGULARITY CRITERIA FOR WEAK SOLUTION TO THE 3D MAGNETOHYDRODYNAMIC EQUATIONS

In this article, regularity criteria for the 3D magnetohydrodynamic equations are investigated. Some sufficient integrability conditions on two components or the gradient of two components of u ＋ B and u - B in Morrey-Campanato spaces are obtained.

GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY

This paper studies the global regularity of 2D incompressible anisotropic magnetomicropolar fluid equations with partial viscosity. Ma [22](Ma L. Nonlinear Anal: Real World Appl, 2018, 40: 95–129) examined the global regularity of the 2D incompressible magnetomicropolar fluid system for 21 anisotropic partial viscosity cases. He proved the global existence of a classical solution for some cases and established the conditional global regularity for some other cases. In this paper, we also investigate the global regularity of 12 cases in [22]and some other new partial viscosity cases. The global regularity is established by providing new regular conditions. Our work improves some results in [22] in this sense of weaker regular criteria.

Optimal control approach to stability criteria on Hill's equations

This paper gives a tutorial on how to prove Lyapunov type criteria by optimal control methods. Firstly, we consider stability criteria on Hill’s equations with nonnegative potential. By optimal control methods developed in 1990s, we obtain several stability criteria including Lyapunov’s criterion, Neǐgauz and Lidskiǐ’s criterion. Secondly, we present stability criteria on Hill’s equations with sign-changing potential in which Brog’s criterion and Krein’s criterion are included.

Interval Criteria for Oscillation of Forced Second-order Half-linear Differential Equations

By employing a generalized Riccati technique and an integral averaging tech-nique, new interval oscillation criteria are established for the forced second-order half-lineardifferential equation [r(t)|x′ (t)|α-1x′ (t)]′ + q(t)|x(t)|α-1x(t) = e(t).

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are HSlder continuous near boundary provided that the scaled mixed Lx,t^p,q-norm of the velocity vector field with 3/p ＋ 2/q 〈 2, 2 〈 q 〈 ∞ is sufficiently small near the boundary. Also, we will investigate that for this 3 2〈3 solution U ∈ Lx,t^p,q with 1 〈 3＋p ＋2/q＋≤3/2, 3 〈 p 〈 ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}（3/q＋2/q- 1）.

New Interval Oscillation Criteria for Forced Second-order Half-linear Differential Equations

This paper discusses a class of forced second-order half-linear differential equations. By using the generalized Riccati technique and the averaging technique, some new interval oscillation criteria are obtained.

Interval Criteria for Oscillation of CertainSecond-order Nonlinear Differential Equations

The main desire of this paper is to present several new interval oscillation criteria related to a generalized Riccati technique for certain second-order nonlinear differential equations.Our results extend some known equations.Finally,several examples illustrate the effectiveness of our results.

REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS

In this article, we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in R3. We obtain the classical blow-up criteria for smooth solutions （u,w, b）, i.e., u ∈ Lq（0, T; LP（R3） for 2/q＋3/P≤ 1with 3〈P≤∞,u∈C（[0,T）;L3（R3））or△u∈Lq（0,T,LP）for 3/2〈P≤∞ satisfying 2/q＋3/P≤ 2. Moreover, our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid. In the end-point case p = ∞, the blow-up criteriacan be extended to more general spaces △u E L1 （0, T; B0∞,∞（R3）.

NONOSCILLATION RESULTS FOR A CLASS OF THIRD ORDER NONLINEAR DIFFERENTIAL EQUATIONS

Sufficient conditions for the nonoscillatory solutions of a class of third order nonlinear differential equations are presented. The results obtained generalize some criteria given by Parhi. In some special cases, some of these results contain weaker conditions.

Oscillation theorems for second-order nonlinear perturbed differential equations with damping

A class of second-order nonlinear damped perturbed differential equations is considered and its oscillation theorems are studied.These theorems are more general and deal with the cases which are not covered by the known criteria.Particularly,these criteria extend and unify some existing results.An example is given to verify the results.

Further Oscillation Results for a Class of Hyperbolic Equations with Continuous Distributed Deviating Arguments

A class of hyperbolic equations with continuous distributed deviating arguments is considered and its oscillation theorems are discussed.These theorems are of higher degree of generality and deal with the cases which are not covered by the known criteria.Particularly,these criteria extend and unify a number of existing results.

A Regularity Criterion Via the Pressure On the Three-dimensional Micropolar Fluid Equations in Besov Space

In this paper,we investigate the regularity criterion via the pressure of weak solutions to the micropolar fluid equations in three dimensions.We obtain that for 0<α<1 if p€L^(2/a)(0,T;B^α∞,∞),then the weak solution(u,ω)is regular on(0,T).

Regularity Criterion of Weak Solutions to the 3D Micropolar Fluid Equations in Besov Space

This paper studies the regularity criterion of weak solutions to the micropolarfluid equations in three dimensions. Let (■u∈L2/2-γ(0,T;B^-γ∞,∞),■ω∈L^2(0,T;B^-1∞,∞)),it is showed that the weak solution(u,ω) is globally regular for the case 0 < γ < 2.

CRITERIA OF FINITE ELEMFNT ALGORITHM FOR A CLASS OF PARABOLIC EQUATION

In finite element analysis of transient temperature field, it is quite notorious that the numerical solution may quite likely oscillate and/or exceed the reasonable scope, which violates the natural law of heat conduction. For this reason, we put forward the concept of lime monotony and spatial monotony, and then derive several sufficient conditions for nionotonic solutions in lime dimension for 3-D passive heal conduction equations with a group of finite difference schemes. For some special boundary conditions and regular element meshes, the lower and upper bounds for can be obtained from those conditions so that reasonable numerical solutions are guaranteed. Spatial monotony is also discussed. Finally, the lumped mass method is analyzed.We creatively give several new criteria for the finite element solutions of a class of parabolic equation represented by heal conduction equation.

非线性三阶时滞微分方程的振动性

利用Riccati变换、不等式技巧和分析性质,研究了一类三阶非线性时滞微分方程解的振动性和渐近性,获得了该类方程振动的充分条件,最后用例子作了验证.

On The Cauchy Problem for A 1D Euler-Alignment System in Besov Spaces

In this paper, we investigate a 1D pressureless Euler-alignment system with a non-local alignment term, describing a kind of self-organizing problem for flocking. As a result, by the transport equation theory and Lagrange coordinate transformation, the local well-posedness of the solutions for the 1D pressureless Euler-alignment in Besov spaces with 1≤p∞ is established. Next, the ill-posedness of the solutions for this model in Besov spaces with 1≤p and is also deduced. Finally, the precise blow-up criteria of the solutions for this system is presented in Besov spaces with 1≤p .

一类带阻尼项的分数阶微分方程的振动性质

通过利用Riccati变换和不等式技巧,获得了一类带阻尼项的非线性分数阶微分方程的振动准则,推广和改进了最近文献的结果,并举例说明。