In this paper, we prove that the full repressilator equations in dimension six undergo a supercritical Hopf bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow-up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The main goal of this article is to study the asymptotic properties and oscillation of the third-order neutral differential equations with discrete and distributed delay. We give several theorems and related examples to illustrate the applicability of these theorems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider an inverse source problem of identification of *F*(*t*) function in the linear parabolic equation *u*_{t} = *u*_{xx} + *F*(*t*) and *u*_{0}(*x*) function as the initial condition from the measured final data and local boundary data. Based on the optimal control framework by Green's function, we construct Fréchet derivative of Tikhonov functional. The stability of the minimizer is established from the necessary condition. The CG algorithm based on the Fréchet derivative is applied to the inverse problem, and results are presented for a test example. Copyright © 2014 John Wiley & Sons, Ltd.

This paper deals with the Cauchy problem for a doubly degenerate parabolic equation with variable coefficient

For the case *λ* + 1 ≥ *N*, one proves that depending on the behavior of the variable coefficient at infinity, the Cauchy problem either possesses the property of finite speed of propagation of perturbation or the support blows up in finite time. This completes a result by Tedeev (A.F.Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Appl. Anal. 86 (2007) 755–782.), which asserts the same result under the condition *λ* + 1 < *N*. Copyright © 2014 John Wiley & Sons, Ltd.

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(*X*) of all self-homeomorphisms of a topological space *X*. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo-distance *d*_{G} associated with a group *G* ⊂ Homeo(*X*), we need to adapt persistent homology and consider *G*-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of *G*. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in *G*-invariant persistent homology with respect to the natural pseudo-distance *d*_{G}. We also show how *G*-invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self-homeomorphisms of a topological space. In this paper, we will assume that the space *X* is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption. Copyright © 2014 John Wiley & Sons, Ltd.

We use the bifurcation method of dynamical systems to study the (2+1)-dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow-up wave solutions, periodic smooth wave solutions, periodic blow-up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow-up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study the existence of multiple positive solutions for a degenerate nonlocal problem on unbounded domain. Using the Ekeland's variational principle combined with the mountain pass theorem, we show that problem admits at least two positive solutions under several different conditions. Copyright © 2014 John Wiley & Sons, Ltd.

Considered herein is a generalized two-component Camassa–Holm system in spatially periodic setting. We first prove two conservation laws; then under proper assumptions on the initial data, we show the precise blow-up scenarios and sufficient conditions guaranteeing the formation of singularities to the solutions of the generalized Camassa–Holm system. Copyright © 2014 John Wiley & Sons, Ltd.

This paper deals with the parabolic–elliptic Keller–Segel system with signal-dependent chemotactic sensitivity function,

under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying *u*_{0} ≥ 0 and . The chemotactic sensitivity function *χ*(*v*) is assumed to satisfy

The global existence of weak solutions in the special case is shown by Biler (*Adv. Math. Sci. Appl. *1999; 9:347–359). Uniform boundedness and blow-up of *radial* solutions are studied by Nagai and Senba (*Adv. Math. Sci. Appl. *1998; 8:145–156). However, the global existence and uniform boundedness of classical *nonradial* solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where *γ* > 0 is a constant depending on Ω and *u*_{0}. Copyright © 2014 John Wiley & Sons, Ltd.

A system of equations for description of the predator–prey relations is considered. The model corresponds to the modified Lotka–Volterra system with logistic growth of the prey and with both predator and prey dispersing by diffusion. The Painlevé analysis of the system of equations is studied. Exact traveling wave solutions are found by means of the *Q*-function method. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we investigate a class of delay Nicholson's blowflies model with a linear harvesting term, new criteria for the existence and convergence dynamics of positive pseudo almost periodic solutions are established by using the fixed point method and the properties of pseudo almost periodic function, together with constructing suitable Lyapunov functionals. The obtained results extend previously known results, and they also partially answer an open problem proposed by L. Berezansky *et al.* Finally, an example with simulation is presented to demonstrate the effectiveness of theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.

We study the large time behavior of non-negative solutions to the nonlinear fractional reaction–diffusion equation *∂*_{t}*u* = − *t*^{σ}( − Δ)^{α ∕ 2}*u* − *h*(*t*)*u*^{p} (*α* ∈ (0,2]) posed on and supplemented with an integrable initial condition, where *σ* ≥ 0, *p* > 1, and *h* : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function *h*, we show that the asymptotic behavior of the mass can be classified along two cases as follows:

if , then there exists

*M*_{ ∞ }∈ (0, ∞ ) such that ;if , then .

Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, a mathematical model for HIV-1 infection with antibody, cytotoxic T-lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection-free and infected steady states is investigated. The basic reproduction number *R*_{0} is identified for the proposed system. If *R*_{0} < 1, then there is an infection-free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for *R*_{0} > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.

We study the ultra-relativistic Euler equations for an ideal gas, which is a system of nonlinear hyperbolic conservation laws. We first analyze the single shocks and rarefaction waves and solve the Riemann problem in a constructive way. Especially, we develop an own parametrization for single shocks, which will be used to derive a new explicit shock interaction formula. This shock interaction formula plays an important role in the study of the ultra-relativistic Euler equations. One application will be presented in this paper, namely, the construction of explicit solutions including shock fronts, which gives an interesting example for the non-backward uniqueness of our hyperbolic system. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study the global existence of classical solutions to the three-dimensional compressible Navier–Stokes equations with a density-dependent viscosity coefficient (*λ* = *λ*(*ρ*)). For the general initial data, which could be either vacuum or non-vacuum, we prove the global existence of classical solutions, under the assumption that the viscosity coefficient *μ* is large enough. Copyright © 2014 John Wiley & Sons, Ltd.

This paper considers a stochastic Gilpin–Ayala model with jumps. First, we show the model that has a unique global positive solution. Then we establish the sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of the solution. The threshold between weak persistence and extinction is obtained. Finally, we make simulations to conform our analytical results. The results show that the jump process can change the properties of the population model significantly. Copyright © 2014 John Wiley & Sons, Ltd.

]]>It is shown in the limit-circle case that system of root functions of the non-self-adjoint maximal dissipative (accumulative) Bessel operator and its perturbation Sturm–Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the maximal dissipative (accumulative) Bessel operators is investigated, and it is proved that system of root functions form a basis (Riesz and Bari bases) in the same Hilbert space. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We present a traffic flow model consisting of a gluing between the Lighthill–Whitham and Richards macroscopic model with a first-order microscopic following the leader model. The basic analytical properties of this model are investigated. Existence and uniqueness are proved, as well as the basic estimates on the dependence of solutions from the initial data. Moreover, numerical integrations show some qualitative features of the model, in particular the transfer of information among regions where the different models are used. Copyright © 2014 John Wiley & Sons, Ltd.

Our aim in this paper is to define dynamic boundary conditions for several sixth-order Cahn–Hilliard systems. We then study the well-posedness and the dissipativity of the systems derived. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study nonlinear wrinkling dynamics of a vesicle in an extensional flow. Motivated by the recent experiments and linear theory on wrinkles of a quasi-spherical membrane, we are interested in examining the linear theory and exploring wrinkling dynamics in a nonlinear regime. We focus on a quasi-circular vesicle in two dimensions and show that the linear analytical results are qualitatively independent of the number of dimensions. Hence, the two-dimensional studies can provide insights into the full three-dimensional problem. We develop a spectral accurate boundary integral method to simulate the nonlinear evolution of surface tension and the nonlinear interactions between flow and membrane morphology. We demonstrate that for a quasi-circular vesicle, the linear theory well predicts the characteristic wavenumber during the wrinkling dynamics. Nonlinear results of an elongated vesicle show that there exist dumbbell-like stationary shapes in weak flows. For strong flows, wrinkles with pronounced amplitudes will form during the evolution. As far as the shape transition is concerned, our simulations are able to capture the main features of wrinkles observed in the experiments. Interestingly, numerical results reveal that, in addition to wrinkling, asymmetric rotation can occur for slightly tilted vesicles. The mathematical theory and numerical results are expected to lead to a better understanding of related problems in biology such as cell wrinkling. Copyright © 2013 John Wiley & Sons, Ltd.

]]>Symmetry group analysis and similarity reduction of nonlinear system of coupled Burger equations in the form of nonlinear partial differential equation are analyzed via symmetry method. The symmetry method has led to similarity reductions of this equation to solvable form to third-order partial differential equation. The infinitesimal, similarity variables, dependent variables, and reduction have been tabulated. The search for solutions of these systems by using the improved tanh method has yielded certain exact solutions expressed by rational functions. Some figures are given to show the properties of the solutions. Copyright © 2013 John Wiley & Sons, Ltd.

]]>Heat transfer of a power-law non-Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power-law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power-law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.

]]>A low order characteristic-nonconforming finite element method is proposed for solving a two-dimensional convection-dominated transport problem. On the basis of the distinguish property of element, that is, the consistency error can be estimated as order *O*(*h*^{2}), one order higher than that of its interpolation error, the superclose result in broken energy norm is derived for the fully discrete scheme. In the process, we use the interpolation operator instead of the so-called elliptic projection, which is an indispensable tool in the traditional finite element analysis. Furthermore, the global superconvergence is obtained by using the interpolated postprocessing technique. Lastly, some numerical experiments are provided to verify our theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.

This paper deals with the problem of finding minimum-norm fixed point of nonexpansive mappings. We present two types of iteration methods (one is implicit, and the other is explicit). We establish strong convergence theorems for both methods. Some applications are given regarding convex optimization problems and split feasibility problems. These results improve some known results existing in the literatures. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, an inverse problem for space-fractional backward diffusion equation, which is highly ill-posed, is considered. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order *α* ∈ (0,2]. We show that such a problem is severely ill-posed, and further present a simplified Tikhonov regularization method to deal with this problem. Convergence estimate is presented under a priori choice of regularization parameter. Numerical experiments are given to illustrate the accuracy and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

This paper is devoted to studying the factorization method applied to the inverse problem of reconstructing a penetrable anisotropic obstacle from far field patterns. We proved the validity of the factorization method. Copyright © 2013 John Wiley & Sons, Ltd.

]]>The dispersionless Kadomtsev–Petviashvili hierarchy is generalized by introducing two new time series *γ*_{n} and *σ*_{k} with two parameters *η*_{n} and *λ*_{k}. By this hierarchy, we obtain the first type, the second type as well as mixed type of dispersionless Kadomtsev–Petviashvili equation with self-consistent sources and their related conservation equations. In addition, the reduction and constrained flow of this new hierarchy are studied. The first type, the second type and the mixed type of dispersionless Korteweg–de Vries equation with self-consistent sources and of dispersionless Boussinesq equation with self-consistent sources are obtained. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, a graph-theoretic approach for checking exponential stability of the system described by neutral stochastic coupled oscillators network with time-varying delayed coupling is obtained. Based on graph theory and Lyapunov stability theory, delay-dependent criteria are deduced to ensure moment exponential stability and almost sure exponential stability of the addressed system, respectively. These criteria can show how coupling topology, time delays, and stochastic perturbations affect exponential stability of such oscillators network. This method may also be applied to other neutral stochastic coupled systems with time delays. Finally, numerical simulations are presented to show the effectiveness of theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

]]>We consider the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid taking into account internal degrees of freedom. We first show that there exists a unique global strong solution when the given initial data are small in some sense. Then, we deduce the optimal decay rates for velocity vector in *L*^{2} − norm and *L*^{p} − norm for *p* > *n*. These decay estimates depend only on the spatial dimension and the decay properties of the heat solution with the same data. Copyright © 2013 John Wiley & Sons, Ltd.

Based on the theory of semi-global piecewise *C*^{2} solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree-like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we establish necessary and sufficient conditions for the solutions of a second-order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, we consider the existence of positive solution for four-point nonlocal boundary value problems of fractional order. By means of some fixed point theorems, some results on the existence and multiplicity of positive solutions are obtained. Furthermore, we provide a representative example to illustrate a possible application of the established results. Copyright © 2013 John Wiley & Sons, Ltd.

]]>The paper deals with the problem of the stability for a model of a population with delayed dependence of the structure. We study an equilibrium age distribution. We also give conditions under which such distribution is exponentially asymptotically stable. The work continues authors’ research which begun in 2009. Copyright © 2013 John Wiley & Sons, Ltd.

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