Modified Boussinesq System with Variable Coefficients： Classical Lie Approach and Exact Solutions

The Lie-group formalism is applied to investigate the symmetries of the modified Boussinesq system with variable coefficients. We derived the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The reduced systems of ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Global Existence and Exponential Decay for a Nonlinear Viscoelastic Equation with Nonlinear Damping

In this paper we investigate a nonlinear viscoelastic equation with nonlinear damping. Global existence of weak solutions and uniform decay of the energy have been established. The Faedo-Galerkin method and the perturbed energy method are employed to obtain the results.     

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 A NONHOMO- GENEOUS NONAUTONOMOUS SECOND- ORDER BOUNDARY VALUE PROBLEM

The existence of multiple positive solutions is studied for a nonlinear nonauto- nomous second-order boundary value problem with nonhomogeneous boundary con- ditions. In order to describe the growth behaviors of nonlinearity on some bounded sets, two height functions are introduced. By considering the integrals of the height functions and applying the Krasnosel＇skii fixed point theorems on a cone, several new results are proved.     

POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH PARAMETER

In this paper, we consider a class of nonlinear fractional differential equation boun- dary value problem with a parameter. By some fixed point theorems, sufficient con- ditions for the existence, nonexistence and multiplicity of positive solutions to the system are obtained. An example is given to illustrate the main results.     

An Image Denoising-Deblurring Adaptive Variational Problem in BV Space

In this paper, we establish the existence and uniqueness of a BV solution to an initial boundary value problem for a nonlinear integro-differential equation, which is related to a denoising and deblurring variational model in image restoration. Several experiments relevant to the restoration model are performed and numerical results are discussed.     

COMBINED EFFECTS IN A SEMILINEAR POLYHARMONIC PROBLEM IN THE UNIT BALL

Let m be a positive integer and B be the unit ball of Rn(n ≥ 2). We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic boundary value problem(-△)mu = a1(x)uα1+ a2(x)uα2, lim|x|→1u(x)(1- |x|)m-1= 0,where α1, α2 ∈(-1, 1) and a1, a2 are two nonnegative measurable functions on B satisfying some appropriate assumptions related to Karamata regular variation theory.     

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.     

A LOCAL ATTRACTOR FOR A NONLINEAR WAVE EQUATION ARISING IN ELASTIC WAVEGUIDE MODEL

This paper studies the asymptotic behavior of solutions to the initial boundary value problem for a nonlinear wave equation arising in an elastic waveguide model. It proves that under rather mild conditions on g the related solution semigroup possesses a local attractor.     

Blow-up vs.Global Finiteness for an Evolution p-Laplace System with Nonlinear Boundary Conditions

In this paper, the authors consider the positive solutions of the system of the evolution p-Laplacian equations
with nonlinear boundary conditions
and the initial data （u0, v0）, where Ω is a bounded domain in Rn with smooth boundary δΩ, p 〉 2, h（·,·） and s（·,· ） are positive C1 functions, nondecreasing in each variable. The authors find conditions on the functions f, g, h, s that prove the global existence or finite time blow-up of positive solutions for every （u0, v0）.     

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.     

REGULARITY OF THE SOLUTIONS FOR NONLINEAR BIHARMONIC EQUATIONS IN RN

The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation
{△^2u ＋ a（x）u = g（x, u）
u∈ H^2（R^N）,
where the condition u∈ H^2（R^N） plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.     

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

In this article,we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients.The authors use a new method introduced by Duzaar and Grotowski,to prove partial regularity for weak solutions,based on a generalization of the technique of harmonic approximation and directly establish the optimal Ho¨lder exponent for the derivative of a weak solution on its regular set.     

GLOBAL EXISTENCE AND EXTINCTION FOR A DEGENERATE NONLINEAR DIFFUSION PROBLEM WITH NONLINEAR GRADIENT TERM AND ABSORPTION

Existence and extinction in finite time of global weak solutions for the problem （P） are proved.     

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.     

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.     

UNSTABLE SOLUTIONS TO A CLASS OF VECTOR DIFFERENTIAL EQUATIONS OF SIXTH ORDER

In this paper, we consider a class of nonlinear vector differential equations of sixth order. By constructing appropriate Lyapunov functions, the non-existence of periodic solutions is established. Moreover, we provide an example to show the feasibility of our results. Our results extend and improve two related results in the previous literature from scalar cases to vectorial cases.     

Traveling Wave Solutions to the Three-Dimensional Nonlinear Viscoelastic System

In this paper, we establish the existence of traveling wave solutions to the nonlinear three-dimensional viscoelastic system exhibiting long range memory. Under certain hypotheses, if the speed of propagation is between the speeds determined by the equilibrium and instantaneous elastic tensors, then the system has nontrivial trav- eling wave solutions. Moreover, the system has only trivial traveling wave solution in some cases.     

EXISTENCE OF MULTIPLE SOLUTIONS TO A FRACTIONAL DIFFERENCE BOUNDARY VALUE PROBLEM WITH PARAMETER

By establishing the corresponding variational framework, and using critical point theory, we give the existence of multiple solutions to a fractional difference boundary value problem with parameter. Under some suitable assumptions we obtain some results which ensure the existence of well precise interval of parameter for which the problem admits multiple solutions.