BLOW-UP RATE OF SOLUTIONS FOR P-LAPLACIAN EQUATION

In this note we consider the blow-up rate of solutions for p-Laplacian equation with nonlinear source, ut = div（｜↓△u｜p-2↓△u）＋uq, （x, t） ∈ RN × （0, T）, N ≥ 1. When q 〉 p - 1, the blow-up rate of solutions is studied.     

Source Type Solutions of a Fourth Order Degenerate Parabolic Equation

In this paper, we study a generalized thin film equation which is relevant to capillary driven flows of thin films of power-law fluids. We prove that the generalized thin film equation in dimension d ≥ 2 has a unique C^1 source type radial self-similar nonnegative solution if 0 〈 n 〈 2p - 1 and has no solution of this type if n ≥ 2p - 1.     

A Nonlinear Diffusion System with Coupled Nonlinear Boundary Flux

This paper studies a nonlinear diffusion system with coupled nonlinear boundary flux and two kinds of inner sources （positive for the first and negative for the second）, where the four nonlinear mechanisms are described by eight nonlinear parameters. The critical exponent of the system is determined by a complete classification of the eight nonlinear parameters, which is represented via the characteristic algebraic system introduced to the problem.     

A MATHEMATICAL MODEL OF ENTERPRISE COMPETITIVE ABILITY AND PERFORMANCE THROUGH EMDEN-FOWLER EQUATION （Ⅱ）

In this paper we work with the ordinary equation u＇＇ - u2 （u ＋ ） = 0 and ob- tain some interesting phenomena concerning, blow-up, blow-up rate, life-span of solutions to those equations.     

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

In this article,we study the initial boundary value problem of generalized Pochhammer-Chree equation u_(tt)-u_(xx)-u_(xxt)-u_(xxtt)=f(u) xx,x ∈Ω,t 0,u(x,0) = u0(x),u t(x,0)=u1(x),x ∈Ω,u(0,t) = u(1,t) = 0,t≥0,where Ω=(0,1).First,we obtain the existence of local W k,p solutions.Then,we prove that,if f(s) ∈ΩC k+1(R) is nondecreasing,f(0) = 0 and |f(u)|≤C1|u| u 0 f(s)ds+C2,u 0(x),u 1(x) ∈ΩW k,p(Ω) ∩ W 1,p 0(Ω),k ≥ 1,1 p ≤∞,then for any T 0 the problem admits a unique solution u(x,t) ∈ W 2,∞(0,T;W k,p(Ω) ∩ W 1,p 0(Ω)).Finally,the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.     

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 R~3.We obtain the classical blow-up criteria for smooth solutions(u,ω,b),i.e.,u ∈ L q(0,T;L p(R 3)) for 2 q + 3 p ≤ 1 with 3p≤∞,u ∈ C([0,T);L 3(R 3)) or u ∈L q(0,T;L p) for 3 2p≤∞ satisfying 2 q + 3p≤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 criteria can be extended to more general spaces u∈ L~1(0,T;B_(∞,∞)~0(R~3)).     

A REMARK ON THE BLOW-UP CRITERION OF STRONG SOLUTIONS AND REGULARITY FOR WEAK SOLUTIONS OF NAVIER-STOKES EQUATIONS

We give blow-up criteria of strong solutions of Navier-Stokes equations with its initial data in Besov spaces and consider the regularity of Leray-Hopf solutions of the equation.     

The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear Partial Differential Equations with Initial Conditions

New developed inverse differential operators incorporated into the semi- analytical treatment of the modified decomposition method （MDM） are used to solve the systems of first and second-order singular nonlinear partial differential equations （PDEs） with initial conditions arising in physics. The new proposed method is called the improved modified decomposition method （IMDM）, and is used to the treatment of a few case study initial-value problems. The results obtained by the IMDM are in full agreement with the existing exact analytical solutions.     

A PRIORI BOUNDS FOR GLOBAL SOLUTIONS OF HIGHER-ORDER SEMILINEAR PARABOLIC PROBLEMS

In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal regularity estimates. If n≥ 10/3m, we also give another proof which does not use maximal regularity estimates.     

Asymptotic Behavior of Solution to Some Models Involving Two Species All with Chemotaxis

This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis：{αp/αt=Dp△（p△lnp/w）,αp/αt=Dq△（q△lnq/w）,αw/αt=βp-δw,p△ln（p/w）.^-n=q△ln（q/w）.^-n=0.We prove that the solution exists globally asβ ≥ 0. Asβ 〈 0, whether the solution exists globally or not depends on the initial data. By function transformation and compari- son, the asymptotical behavior of the solution is studied.     

NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.     

SOLUTIONS TO DISCRETE MULTIPARAMETER PERIODIC BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN VIA CRITICAL POINT THEORY

In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [G. Bonanno and P. Candito, Appl.Anal., 88(4)(2009), pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.     

Non-simultaneous Blow-up Criteria for Localized Parabolic Equations

This paper deals with blow-up solutions for parabolic equations coupled via localized exponential sources, subject to homogeneous Dirichlet boundary con- ditions. The criteria are proposed to identify simultaneous and non-simultaneous blow-up solutions. The related classification for the four nonlinear parameters in the model is optimal and complete.     

ASYMPTOTIC SOLUTION TO SINGULARLY PERTURBED DELAYED EQUATION FOR PREDATOR-PREY MODEL

A kind of singularly perturbed delayed equation for predator-prey system is consi- dered. According to the boundary layer function method and the singular perturbation theory, we construct an asymptotic expansion of the solution to the original equation.     

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

Let B1 ■ RNbe a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:-div(|▽u|p-2▽u) = |x|s|u|p*(s)-2u + λ|x|t|u|p-2u, x ∈ B1,u|■B1= 0,where t, s -p, 2 ≤ p N, p*(s) =(N+s)p N-pand λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N p(p- 1)t + p(p2- p + 1) and λ∈(0, λ1,t), where λ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p) min{1,p+t p+s}+p2p-(p-1) min{1,p+t p+s}and λ 0 is small.     

SINGULAR INTEGRAL DIFFERENTIAL EQUATIONS WITH BOTH CAUCHY KERNEL AND CONVOLUTION KERNEL AND RIEMANN BOUNDARY VALUE PROBLEM

In this paper, we set up and discuss a kind of singular integral differential equation with convolution kernel and Canchy kernel. By Fourier transform and some lemmas, we turn this class of equations into Riemann boundary value problems, and obtain the general solution and the condition of solvability in class {0}.     

Asymptotically Self-Similar Global Solutions for a Higher-Order Semilinear Parabolic System

In this paper, we study the higher-order semilinear parabolic system{ut＋（-△）^mu=a｜v｜^p-1v,（t,x）∈R^1＋×R^N,vt＋（-△）^mv=b｜u｜^q-1u,（t,x）∈R^1＋×R^N,u（0,x）=φ（x）,v（0,x）=ψ（x）,x∈R^N, where m, p,q 〉 1, a,b ∈R. We prove that the global existence of mild solutions for small initial data with respect to certain norms. Some of these solutions are proved to be asymptotically self-similar.     

Morrey Estimates for Nondivergence Parabolic Operators with Discontinuous Coefficients on Homogeneous Groups

In this paper, by establishing the boundedness of singular integral operators with variable kernels and their commutators with BMO functions on Morrey spaces of homogeneous groups, we prove a local a prior estimate in Sobolev-Morrey space for solutions to the nondivergence parabolic equation with discontinuous coefficients.     

BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter α goes to zero.     

POSITIVE SOLUTIONS TO AN INTEGRAL BOUNDARY VALUE PROBLEM WITH DELAY

In this paper, we study a second order integral boundary value problem with delay. By the Krasnoselskii fixed point theorem, we obtain sufficient conditions for the existence of at least one or two positive solutions to the problem.