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In the present paper, we study the problem of multiple non semi-trivial solutions for the following systems of Kirchhoff-type equations with discontinuous nonlinearities (1.1) where F∈C1(RN×R+×R+,R),V∈C(RN,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi-trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we propose a discrete viral infection model with a general incidence rate. The discrete model is derived from a continuous case by using a 'mixed' Euler method, which is a mixture of both forward and backward Euler methods. We prove that the mixed Euler method preserves the qualitative properties of the corresponding continuous system, such as positivity, boundedness, and global behaviors of solutions. Furthermore, the model and mathematical results presented in another previous study are extended and generalized. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove the existence and uniqueness of the local mild solution to the Cauchy problem of the n-dimensional (n≥3) Wigner–Poisson–BGK equation in the space of some integrable functions whose inverse Fourier transform are integrable. The main difficulties in establishing mild solution are to derive the boundedness and locally Lipschitz properties of the appropriate nonlinear terms in the Wiener algebra. Copyright © 2015 John Wiley & Sons, Ltd.

C. Miao In this paper, we are concerned with the 1D Cauchy problem of the compressible Navier–Stokes equations with the viscosity μ(ρ) = 1+ρβ(β≥0). The initial density can be arbitrarily large and keep a non-vacuum state at far fields. We will establish the global existence of the classical solution for 0≤β < γ via a priori estimates when the initial density contains vacuum in interior interval or is away from the vacuum. We will show that the solution will not develop vacuum in any finite time if the initial density is away from the vacuum. To study the well-posedness of the problem, it is crucial to obtain the upper bound of the density. Some new weighted estimates are applied to obtain our main results. Copyright © 2015 John Wiley & Sons, Ltd.

In thiswork,we present two new(3+1)-dimensional nonlinear equationswith Korteweg-de Vries equation constituting its main part. We show that the dispersive relation is distinct for each model, whereas the phase shift remains the same. We determine multiple solitons solutions, with distinct physical structures, for each established equation. The architectures of the simplified Hirota's method is implemented in this paper. The constraint conditions that fall out which must remain valid in order for themultiple solitons to exist are derived.Copyright © 2015 John Wiley & Sons, Ltd.

This paper studies the Neimark–Sacker bifurcation of a diffusive food-limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank-Nicolson difference scheme, and nonstandard finite-difference scheme. The existence of Neimark-Sacker bifurcation at the equilibrium is obtained. Our results show that Crank-Nicolson and nonstandard finite-difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.

The purpose of this paper is to give a proof of global existence of solutions for Gierer–Meinhardt systems with homogeneous Neumann boundary conditions. Our technique is based on Lyapunov functional argument that yields the uniform boundedness of solutions. The asymptotic behaviour of the solutions under suitable conditions is also studied. Moreover, under reasonable conditions on the exponents of the nonlinear term, we show the blow up in finite time of the solutions for the considered system. These results are valid for any positive initial data in , without any differentiability conditions. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study a class of integral boundary value problem for fractional order impulsive differential equations, where both the nonlinearity and the impulsive terms contain the fractional order derivatives. By using fixed-point theorems, the existence results of solution for the boundary value problem are established. Finally, some examples are presented to illustrate the existence results. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage-to-limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional-order Kaup–Kupershmidt (KK) equation. Two-dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time-fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion: where χ > 0, ζ > 0, αi>0, βi>0 (i = 1,2) and f(s)≤κ − μsτ. In two-space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any μ > 0. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a new numerical method for solving the fractional Bagley-Torvik equation is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the initial and boundary value problems for the fractional Bagley-Torvik differential equation to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.

A fluid–particles system of the compressible Navier-Stokes equations and Vlasov-Fokker-Planck equation (including the case of Vlasov equation) in three-dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, by using the variational method and the critical point theorem because of Brezis and Nirenberg, we investigate the existence of solutions to a class of second-order impulsive Hamiltonian system. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we present a new coupled modified (1 + 1)-dimensional Toda equation of BKP type (Kadomtsev-Petviashvilli equation of B-type), which is a reduction of the (2 + 1)-dimensional Toda equation. Two-soliton and three-soliton solutions to the coupled system are derived. Furthermore, the N-soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two-soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider the nonlinear oscillation of the following second-order neutral delay dynamic equations with distributed delay on a time scale , where Z(t) = x(t) + p(t)x(τ(t)),α,β > 0 are constants. By using some new techniques, we obtain oscillation criteria for the equation when β > α,β = α, and β < α, respectively. Those results established here complete and develop the oscillation criteria in the literature. Also, our main results are illustrated with some examples. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we establish sufficient conditions for the global relative controllability of nonlinear neutral fractional Volterra integro-differential systems with distributed delays in control. The results are obtained by using the Mittag– Leffler functions and the Schauder fixed-point theorem. Examples are presented to illustrate the results. Copyright © 2015 John Wiley & Sons, Ltd.

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of the central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure that reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove the Saint-Venant compatibility conditions in Lp for p∈(1,+∞), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in Lp are provided. We also use the Helmholtz decomposition in Lp to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (p = 2), to the full range p∈(1,+∞). Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.

The paper deals with the oscillation analysis of numerical solution in the θ-methods for differential equations with piecewise constant arguments of advanced type. The conditions of the oscillation for the θ-method are obtained. It is proved that the oscillation of the analytic solution is preserved by the θ- method. Some numerical experiments are given. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the following coupled chemotaxis system with homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn(n≥2) with smooth boundary, where λ, χ1, χ2, μ1, μ2, a1, a2, b1, and b2 are supposed to be positive and τ = 0,1. In the case τ = 0, based on some energy estimates for both u and v, it is shown that for any parameters, the system possesses a unique globally bounded solution if n = 2. Moreover, when τ = 1, relying on a comparison principle, for a range of parameters, the existence of a unique global bounded classical solution of problem is established for any n≥2 if Ω is convex. Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with exact controllability of bilinear heat equation. Namely, given the initial state, we would like to provide a class of target states that can be achieved through the heat equation at a finite time by applying multiplicative controls. For this end, an explicit control strategy is constructed. Simulations are provided. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the Navier–Stokes equations with Navier boundary conditions in a bounded domain of . We prove the existence of weak solution in W1,p(Ω) × Lp(Ω) for by using fixed point theorem for the case . Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove two blow-up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (−Δ)αu in the space Rn,n=2,3 and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in [0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided ∇u∈L10,T;B∞,∞0. For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if (∇u,∇F)∈L10,T;B∞,∞0, which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the internal exact controllability of a generalized Bresse system with variable coefficients, which the controls functions acts in an arbitrarily small subinterval (l1,l2) of (0,L). Our computation suggests a minimal time control and a region where the controls are more effective. The variable coefficients can be viewed as a generalization of Laplacian operator. The main result is obtained by applying Hilbert Uniqueness Method proposed by Lions, without using the Holmgren's uniqueness theorem or the hypothesis of equal-speed waves of propagation. Copyright © 2015 John Wiley & Sons, Ltd.

Criteria for the bifurcation of small solutions of an equation F(lambda,u) = 0 from a line {(lambda,0): lambda is an element of R} of trivial solutions are usually based on properties of the DuF(lambda,0) at the trivial solutions, where the partial derivative is taken in the sense of Frechet. When this derivative only exists in some weaker sense, the situation charges considerably and much remains to be carried out to understand the conditions under which bifurcation takes place. This paper summarizes and extends one direction of research in this direction, where only Hadamard differentiability is required. In addition to presenting the abstract results, their application to the study of bound states of a class of nonlinear elliptic equations on R-N is also examined. Copyright (c) 2015 John Wiley & Sons, Ltd.

In this paper, we investigate the vacuum free boundary problem of a compressible Navier–Stokes–Poisson system with density-dependent viscosity. By introducing Eulerian and Lagrange energy, we obtain a local in time well-posedness of the strong solution to the Navier–Stokes–Poisson system in a spherically symmetric case. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider the coupled Kirchhoff-type system where e is a small positive parameter and ai>0, bi≥0 are constants for i = 1,2, P,Q are positive continuous potentials satisfying some conditions. Using minimax theorem and the Ljusternik-Schnirelmann category theory, we obtain the existence and multiplicity results of ground state solutions for the aforementioned system as e > 0 small enough. Moreover, the concentration phenomena of solutions is also explored. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study behavior of the solution of the following max-type difference equation system: where , the parameter A is positive real number, and the initial values x0,y0 are positive real numbers. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the nonlinear Schrodinger–Maxwell system where the potential V and the primitive of g are allowed to be sign-changing, and g is local superlinear. Under some simple assumptions on V,Q and g, we establish some existence criteria to guarantee that the aforementioned system has at least one nontrivial solution or infinitely many nontrivial solutions by using critical point theory. Recent results in the literature are generalized and significantly improved. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider a quite general class of reaction-diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time-scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo-inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation. The local well-posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces Hs) by using Littlewood-Paley decomposition and transport equation theory. Moreover, the local well-posedness in critical case (with ) is considered.

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one-dimensional nonlinear sine-Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth-order for discretizing the spatial derivative and the standard second-order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V-cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one-dimensional sine-Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider the nonlocal non-autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non-negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the asymptotic behavior of the decreasing energy solution ue to a p-Ginzburg–Landau system with the initial-boundary data for p > 4/3. It is proved that the zeros of ue in the parabolic domain G × (0,T] are located near finite lines {ai}×(0,T]. In particular, all the zeros converge to these lines when the parameter e goes to zero. In addition, the author also considers the uniform energy estimation on a domain far away from the zeros. At last, the Holder convergence of ue to a heat flow of p-harmonic map on this domain is proved when p > 2. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one-periodic and two-periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, the Backlund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, a time-delayed free boundary problem for tumor growth under the action of external inhibitors is studied. It is assumed that the process of proliferation is delayed compared with apoptosis. By Lp theory of parabolic equations, the Banach fixed point theorem and the continuation theorem, the existence and uniqueness of a global solution is proved. The asymptotic behavior of the solution is also studied. The proof uses the comparison principle and the iteration method. Copyright © 2014 John Wiley & Sons, Ltd.

The paper deals with the existence and uniqueness of classical solutions of the homogeneous Neumann problem for a class of parabolic–hyperbolic system of partial differential equations in n dimensions. The problem arises from a model of the diffusion of N species of radioactive isotopes of the same element. Copyright © 2014 John Wiley & Sons, Ltd.

The paper deals with the two-dimensional problem of the elastic equilibrium of a body containing a thin rigid inclusion. The rigid inclusion is considered to be soldered into elastic medium. The volume forces act on the body; at the external boundary the body is fixed. The first and second variations of the solution with respect to the shape of the domain are calculated. As an illustrative application of the results, the optimal shape design problem is considered. The necessary and sufficient conditions for local optimality are written. Copyright © 2014 John Wiley & Sons, Ltd.

This paper concerns the 3D Navier-Stokes equations and prove an almost Serrin-type regularity criterion in terms of one directional derivative of the pressure. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov–Morrey spaces. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we present a novel numerical scheme to solve a 1D, one-phase extended Stefan problem with fractional Caputo derivative with respect to time. The proposed method is based on a suitable choice of the new space coordinate for the subdiffusion equation and extends the front-fixing method to the subdiffusion case. In the final part, examples of numerical results are discussed. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study the numerical solution to time-fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time-fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L2-based Sobolev spaces. A useful intermediary result is the well-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L2-based Sobolev spaces. We obtain this well-posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well-posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B-vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.

In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The a priori and a posteriori rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.