We consider a semilinear wave equation with nonlinear damping in the whole space . Local-in-time existence and uniqueness results are obtained in the class of Bessel-potential spaces . Copyright © 2017 John Wiley & Sons, Ltd.

]]>Ren and Zeng (2013) introduced a new kind of *q*-Bernstein–Schurer operators and studied some approximation properties. Acu *et al*. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using *q*-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's *K*-functional. Next, we introduce the bivariate case of *q*-Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's *K*-functional. Finally, we define the generalized Boolean sum operators of the *q*-Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

This paper presents an operatorial model based on fermionic operators for the description of the dynamics of political parties affected by turncoat-like behaviors. By observing the political landscape in place in Italy over the last years, appropriate macro-groups have been identified on the basis of the behavior of politicians in terms of disloyal attitude as well as openness towards accepting chameleons from other parties. Once introduced, a time-dependent number-like operator for each physical observable relevant for the description of the political environment, the analysis of the party system dynamics is carried out by combining the action of a quadratic Hamiltonian operator with certain rules acting periodically on the system in such a way that the parameters entering the model are repeatedly changed so as to express a sort of dependence of them upon the variations of the mean values of the observables. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, we discuss the integral representation of quaternionic harmonic functions in the half space with the general boundary condition. Next, we derive a lower bound from an upper one for quaternionic harmonic functions. These results generalize some of the classic results from the case of plane to the case of noncommutative quaterninionic half space. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, multi-switching combination–combination synchronization scheme has been investigated between a class of four non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional-order Lü and Rössler chaotic systems. In multi-switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional-order chaotic systems, the multi-switching combination–combination synchronization of four fractional-order non-identical systems has been investigated. For the synchronization of four non-identical fractional-order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this work, we integrate both density-dependent diffusion process and Beddington–DeAngelis functional response into virus infection models to consider their combined effects on viral infection and its control. We perform global analysis by constructing Lyapunov functions and prove that the system is well posed. We investigated the viral dynamics for scenarios of single-strain and multi-strain viruses and find that, for the multi-strain model, if the basic reproduction number for all viral strains is greater than 1, then each strain persists in the host. Our investigation indicates that treating a patient using only a single type of therapy may cause competitive exclusion, which is disadvantageous to the patient's health. For patients infected with several viral strains, the combination of several therapies is a better choice. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A five-dimensional ordinary differential equation model describing the transmission of *Toxoplamosis gondii* disease between human and cat populations is studied in this paper. Self-diffusion modeling the spatial dynamics of the *T. gondii* disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans-critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease-free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three-dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in *B**M**O* norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO-type estimate for the Lam
system and a variant of the Brezis-Waigner's inequality play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,*T*], for a class of higher-order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.

In this work, we study the approximation of traveling wave solutions propagated at minumum speeds *c*_{0}(*h*) of the delayed Nicholson's blowflies equation:

In order to do that, we construct a subsolution and a super solution to (∗). Also, through that construction, an alternative proof of the existence of traveling waves moving at minimum speed is given. Our basic hypothesis is that *p*/*δ*∈(1,*e*] and then, the monostability of the reaction term. Copyright © 2017 John Wiley & Sons, Ltd.

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A novel second-order two-scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non-periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non-periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell-centered primary unknowns and cell edge-based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell-centered one. The derivation of the scheme satisfies the linearity-preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge-based linearity-preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The purpose of this paper is to introduce a family of *q*-Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well-known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's *K*-functional, and the second-order modulus of continuity. Furthermore, we obtain the approximation results for bivariate *q*-Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.

We discuss bound and anti-bound states for 2×2 matrix Schrödinger operator. We analyze the Fredholm determinant for Hamiltonians that can be represented in a multi-channel framework. Our analysis covers the whole and the half-line problems. We obtain some results on counting anti-bound states between successive bound states. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible-Infective-Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number . If , then the disease-free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by-product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, we establish sufficient conditions for the regularity of solutions of 3D MHD equations in the framework of the anisotropic Lebesgue spaces. In particular, we obtain the anisotropic regularity criterion via partial derivatives, and it is a generalization of the some previous results. Besides, the anisotropic integrability regularity criteria in terms of the magnetic field and the third component of the velocity field are also investigated. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two-dimensional sphere close to its poles. This equation is the counterpart of the well-studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is *C*^{1} smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.

This paper considers the two-dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non-selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non-selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the dynamics of a Nicholson's blowflies equation with state-dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state-dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to help in designing experiments to develop novel therapeutic strategies against cancer. In this paper, by using theory of functional and ordinary differential equations, we study the existence and stability of nonlinear growth kinetics of breast cancer stem cells. First, we provide a sufficient condition for the existence and uniqueness of the solution for nonlinear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a nontrivial steady-state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions that allow for this criteria to be satisfied. Next, we apply these theorems to a special case of the system of functional differential equations that has been used to model nonlinear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate the qualitative behavior of a host-parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole-placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper addresses the analysis of a time noise-driven Allen–Cahn equation modelling the evolution of damage in continuum media in the presence of stochastic dynamics. The nonlinear character of the equation is mainly due to a multivoque maximal monotone operator representing a constraint on the damage variable, which is forced to take physically admissible values. By a Yosida approximation and a time-discretization procedure, we prove a result of global-in-time existence and uniqueness of the solution to the stochastic problem. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We analyze some fourth-order partial differential equations that model the ‘propagation of hexagonal patterns’ and the ‘microphase separation of di-block copolymers’. The underlying invariance properties and conservation laws of the models and related partial differential equations are studied. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider second-order ordinary differential equations with discontinuous right-hand side. We analyze the concept of solution of this kind of equations and determine analytical conditions that are satisfied by typical solutions. Moreover, the existence and uniqueness of solutions and sliding solutions are studied. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, by using the fractional calculus, measure of noncompactness, and the Mönch's fixed point theorem, we investigate the controllability results for fractional neutral integrodifferential equations with nonlocal conditions in Banach spaces. In the end, we give an example to illustrate the applications of the abstract conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we will firstly extend the results about Jiu, Wang, and Xin (JDE, 2015, 259, 2981–3003). We prove that any smooth solution of compressible fluid will blow up without any restriction about the specific heat ratio *γ*. Then we prove the blow-up of smooth solution of compressible Navier–Stokes equations in half space with Navier-slip boundary. The main ideal is constructing the differential inequality. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, a computational technique based on the pseudo-spectral method is presented for the solution of the optimal control problem constrained with elliptic variational inequality. In fact, our aim in this paper is to present a direct approach for this class of optimal control problems. By using the pseudo-spectral method, the infinite dimensional mathematical programming with equilibrium constraint, which can be an equivalent form of the considered problem, is converted to a finite dimensional mathematical programming with complementarity constraint. Then, the finite dimensional problem can be solved by the well-developed methods. Finally, numerical examples are presented to show the validity and efficiency of the technique. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The paper deals with the asymptotic formulation and justification of a mechanical model for a dynamic piezoelastic shallow shell in Cartesian coordinates. Starting from the three-dimensional dynamic piezoelastic problem and by an asymptotic approach, the authors study the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. In order to obtain a nontrivial limit problem by asymptotic analysis, we need different scalings on the mass density. The authors show that the transverse mechanical displacement field coupled with the in-plane components solves an problem with new piezoelectric characteristics and also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The construction of modified two-step hybrid methods for the numerical solution of second-order initial value problems with periodic or oscillatory behavior is considered. The coefficients of the new methods depend on the frequency of each problem so that the harmonic oscillator is integrated exactly. Numerical experiments indicate that the new methods are more efficient than existing methods with constant or variable coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper deals with asymptotic behavior for blow-up solutions to time-weighted reaction–diffusion equations *u*_{t}=Δ*u*+*e*^{αt}*v*^{p} and *v*_{t}=Δ*v*+*e*^{βt}*u*^{q}, subject to homogeneous Dirichlet boundary. The time-weighted blow-up rates are defined and obtained by ways of the scaling or auxiliary-function methods for all *α*,
. Aiding by key inequalities between components of solutions, we give lower pointwise blow-up profiles for single-point blow-up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow-up rates and new blow-up versus global existence criteria are obtained. Time-weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow-up rates, but they do not limit the order of time-weighted blow-up rates and pointwise profile near blow-up time. Copyright © 2017 John Wiley & Sons, Ltd.

Quasi-periodic piecewise analytic solutions, without poles, are found for the local antiplane-strain problems. Such problems arise from applying the asymptotic homogenization method to an elastic problem in a parallel fiber-reinforced periodic composite that presents an imperfect contact of spring type between the fiber and the matrix. Our methodology consists of rewriting the contact conditions in a complex appropriate form that allow us to use the elliptic integrals of Cauchy type. Several general conditions are assumed including that the fibers are disposed of arbitrary manner in the unit cell, that all fibers present imperfect contact with different constants of imperfection, and that their cross section is smooth closed arbitrary curves. Finally, we obtain a family of piecewise analytic solutions for the local antiplane-strain problems that depend of a real parameter. When we vary this parameter, it is possible to improve classic bounds for the effective coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface with the boundary is investigated. The main objective is to trace what happens in Γ-limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ-limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space . Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider the inviscid limit for the periodic solutions to Navier–Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier–Stokes equation is independent of viscosity, and that the solutions of the Navier–Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover, the convergence rate in Gevrey class is presented. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The oscillation of solutions of the *n*th-order delay differential equation

was studied in [S. R. Grace and A. Zafer, Math. Meth. Appl. Sci. 2016, 39 1150–1158] when *n* is even and the *n* odd case has been referred to as an interesting open problem. In the present work, our primary aim is to address this situation. Our method of the proof that is quite different from the aforementioned study is essentially new. We introduce *V*_{n−1}-type solutions and use comparisons with first-order oscillatory and second-order nonoscillatory equations. Examples are given to illustrate the main results. Copyright © 2017 John Wiley & Sons, Ltd.

This paper is concerned with the Cauchy problem of the two-dimensional Euler–Boussinesq system with stratification effects. We obtain the global existence of a unique solution to this system without assumptions of small initial data in Sobolev spaces. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider smooth compactly supported solution to the classical Vlasov–Poisson system in three space dimensions in the electrostatic case. An estimate on velocities is derived, showing a growth rate at most like the power 1/8 of the time variable. As a consequence, a better decay estimate is obtained for the electric field in the norm. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we prove the existence of ground state sign-changing solutions for the following class of elliptic equation:

where
, and *K*(*x*) are positive continuous functions. Firstly, we obtain one ground state sign-changing solution *u*_{b} by using some new analytical skills and non-Nehari manifold method. Furthermore, the energy of *u*_{b} is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of *u*_{b} as the parameter *b*↘0. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider an abstract wave equation in the presence of memory. The viscoelastic kernel *g*(*t*) is subject to a general assumption
, where the function *H*(·)∈*C*^{1}(*R*^{+}) is positive, increasing and convex with *H*(0)=0. We give the decay result as a solution to a given nonlinear dissipative ODE governed by the function *H*(*s*). Copyright © 2017 John Wiley & Sons, Ltd.

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate random forcing. The existence and pathwise uniqueness of strong solutions with *H*^{1}-initial data is established, and then the existence of an invariant measure for the Feller semigroup is shown by Krylov–Bogoliubov theorem. Because of the coupled items in the stochastic Ginzburg–Landau–Newell equations, the higher order momentum estimates can be only obtained in the *L*^{2}-norm. We show the ergodicity of invariant measure for the transition semigroup by asymptotically strong Feller property and the support property. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we study the global well-posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth-order wave equation [[EQUATION1]] where
if *d*⩽4, and
if *d*⩾5. The main strategy we use in this paper is concentration-compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. Copyright © 2017 John Wiley & Sons, Ltd.

Optical vortices as topological objects exist ubiquitously in nature. In this paper, we use the principle of variational method and mountain pass lemma to develop some existence theorems for the stationary vortex wave solution of a coupled nonlinear Schrödinger equations, which describe the possibility of effective waveguiding of a weak probe beam via the cross-phase modulation-type interaction. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Additionally, as demanded by beam confinement, we prove the exponential decay of the soliton amplitude at infinity. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a class of singular quasilinear Schrödinger equations of the form

where
are given functions, *N*⩾3,*λ* is a positive constant,
. By using variational methods together with concentration–compactness principle, we prove the existence of positive solutions of the aforementioned equations under suitable conditions on *V*(*x*) and *K*(*x*). Copyright © 2017 John Wiley & Sons, Ltd.

A new mathematical model included an exposed compartment is established in consideration of incubation period of schistosoma in human body. The basic reproduction number is calculated to illustrate the threshold of disease outbreak. The existence of the disease free equilibrium and the endemic equilibrium are proved. Studies about stability behaviors of the model are exploited. Moreover, control measure assessments are investigated in order to seek out effective control interventions for anti-schistosomiasis. Then, the corresponding optimal control problem according to the model is presented and solved. Theoretical analyses and numerical simulations induce several prevention and control strategies for anti-schistosomiasis. At last, a discussion is provided about our results and further work. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We prove the existence of weak solutions to a one-dimensional initial-boundary value problem for a model system of partial differential equations, which consists of a sub-system of linear elasticity and a nonlinear non-uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function in that work by . The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the variable-coefficient fractional diffusion equations with two-sided fractional derivative. By introducing an intermediate variable, we propose a mixed-type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over
. On the basis of the formulation, we develop a mixed-type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz-like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from *O*(*N*^{2}) to *O*(*N*) and the computing cost from *O*(*N*^{3}) to *O*(*N*log*N*). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we prove the existence and uniqueness of a solution for a class of backward stochastic differential equations driven by *G*-Brownian motion with subdifferential operator by means of the Moreau–Yosida approximation method. Moreover, we give a probabilistic interpretation for the viscosity solutions of a kind of nonlinear variational inequalities. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the non-autonomous Navier–Stokes equations with discontinuous initial data. We prove the global existence of solutions, the decay rate of density, and the equilibrium state of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider interactions of smooth and discontinuous germs as generalized integrations over non-rectifiable paths with applications in theory of boundary value problems of complex analysis. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the initial-boundary value problem for a model of motion of aqueous polymer solutions in a bounded three-dimensional domain subject to the Navier slip boundary condition. We construct a global (in time) weak solution to this problem. Moreover, we establish some uniqueness results, assuming additional regularity for weak solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the nonexistence result for the weighted Lane–Emden equation:

- (0.1)

and the weighted Lane–Emden equation with nonlinear Neumann boundary condition:

- (0.2)

where *f*(|*x*|) and *g*(|*x*|) are the radial and continuously differential functions,
is an upper half space in
, and
. Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems (0.1) and (0.2) under appropriate assumptions on *f*(|*x*|) and *g*(|*x*|). Copyright © 2017 John Wiley & Sons, Ltd.

Synchronization of complex networks with time-varying coupling matrices is studied in this paper. Two kinds of time-varying coupling are taken into account. One is the time-varying inner coupling in the node state space and the other is the time-varying outer coupling in the network topology space. By respectively setting linear controllers and adaptive controllers, time-varying complex networks can be synchronized to a desired state. Meanwhile, different influences of the control parameters of linear controllers and adaptive controllers on the synchronization have also been investigated. Based on the Lyapunov function theory, we construct appropriate positive-definite functions, and several sufficient synchronization criteria are obtained. Numerical simulations further illustrate the effectiveness of conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A discrete multi-group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The existence of one non-trivial solution for a second-order impulsive differential inclusion is established. More precisely, a recent critical point result is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non-trivial anti-periodic solution. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The seasonality of conception for populations of the past using no contraception has remained a *terra incognita*. First, the influence of that of marriages on the seasonality of births is highlighted, taking into account the different stages in women's reproductive lives and the presence of successive cohorts of unequal size. Second, the age-dependent and time-dependent monthly distribution of conception is disentangled from monthly marriage and birth time series by means of stochastic optimization under a Leslie recursion with time-varying and age-varying probability of conception. The application to Armenian-Gregorians in the Don Army Territory (South Russia) from 1889 to 1912 reveals strong consistency between reconstructed conception, mean age at marriage, and fertility time series. Copyright © 2017 John Wiley & Sons, Ltd.

We present an analytic approach to solve a degenerate parabolic problem associated with the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second-order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half-plane. Copyright © 2017 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>In this work, a modified Leslie–Gower predator–prey model is analyzed, considering an alternative food for the predator and a ratio-dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin **(****0****,****0****)**. Given the importance of the origin **(****0****,****0****)** as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non-hyperbolic equilibrium point **(****0****,****0****)** in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio-dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non-negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright © 2017 John Wiley & Sons, Ltd.

In this study, we solve an inverse nodal problem for *p*-Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions *r(x)* and *q(x)* in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.

The initial-boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analyzed by the method of the test functions. This analytical *a priori* information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the stochastic stability under small Gauss type random excitation is investigated theoretically and numerically. When *p* is larger than 0, the *p*-moment stability theorem of stochastic models is proved by Lyapunov method, Ito isometry formula, matrix theory and so on. Then the application of *p*-moment such as *k*-order moment of the origin and *k*-order moment of the center is introduced and analyzed. Finally, *p*-moment stability of the power system is verified through the simulation example of a one machine and infinite bus system. Copyright © 2016 John Wiley & Sons, Ltd.

The main work is related to show the existence and uniqueness of solution for the fractional impulsive differential equation of order *α*∈(1,2) with an integral boundary condition and finite delay. Using the application of the Banach and Sadovaskii fixed-point theorems, we obtain the main results. An example is presented at the end to verify the results of the paper. Copyright © 2016 John Wiley & Sons, Ltd.

The quaternion Fourier transform (QFT) is one of the key tools in studying color image processing. Indeed, a deep understanding of the QFT has created the color images to be transformed as whole, rather than as color separated component. In addition, understanding the QFT paves the way for understanding other integral transform, such as the quaternion fractional Fourier transform, quaternion linear canonical transform, and quaternion Wigner–Ville distribution. The aim of this paper is twofold: first to provide some of the theoretical background regarding the quaternion bound variation function. We then apply it to derive the quaternion Fourier and linear canonical inversion formulas. Secondly, to provide some in tuition for how the quaternion Fourier and linear canonical inversion theorems work on the absolutely integrable function space. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The shrinkage of fossil fuel resources motivates many countries to search alternative energy sources. *Jatropha curcas* is a small drought-resistant shrub from whose seeds a high grade fuel biodiesel can be produced. It is cultivated in many tropical countries including India. However, the plant is affected by the mosaic virus (*Begomovirus*) through infected white-flies (*Bemisia tabaci*) which causes mosaic disease. Disease control is an important factor to obtain healthy crop but in agricultural practice, farming awareness is equally important. Here, we propose a mathematical model for media campaigns for raising awareness among people to protect this plant in small plots and control disease. In order to archive high crop yield, we consider the awareness campaign to be arranged in impulsive way to make people aware from infected white-flies to protect Jatropha plants from mosaic virus. The study reveals that the spread of mosaic disease can be contained or even eradicated by the awareness campaigns. To attain an effective eradication, awareness campaign should be implemented at sufficiently short time intervals. Copyright © 2016 John Wiley & Sons, Ltd.

Our aim in this paper is to study the well-posedness of a singular reaction-diffusion equation which is related with brain lactate kinetics, when spatial diffusion is taken into account. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time-fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill-posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the conductivity problem with piecewise-constant conductivity and Robin-type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non-monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius-type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first-order and second-order shape derivative of the proposed cost functional, which is performed rigorously by means of shape-Lagrangian formulation without using the shape sensitivity of the states variables. © 2016 The Author. *Mathematical Methods in the Applied Sciences* Published by John Wiley & Sons Ltd.

A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Our interest is to quantify the spread of an infective process with latency period and generic incidence rate that takes place in a finite and homogeneous population.

Within a stochastic framework, two random variables are defined to describe the variations of the number of secondary cases produced by an index case inside of a closed population. Computational algorithms are presented in order to characterize both random variables. Finally, theoretical and algorithmic results are illustrated by several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body.

The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we consider a nonlinear coupled wave equations with initial-boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow-up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we prove the local-in-time existence and a blow-up criterion of solutions in the Besov spaces for the Euler-*α* equations of inviscid incompressible fluid flows in
. We also establish the convergence rate of the solutions of the Euler-*α* equations to the corresponding solutions of the Euler equations as the regularization parameter *α* approaches 0 in
. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we derive five new versions of Landau-type theorems for biharmonic mappings of unit disk. Moreover, we prove that all five results are sharp when the bounds are equal to 1. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider a uniform elliptic nonlocal operator

- (1)

which is a weighted form of fractional Laplacian. We firstly establish three maximum principles for antisymmetric functions with respect to the nonlocal operator. Then, we obtain symmetry, monotonicity, and nonexistence of solutions to some semilinear equations involving the operator
on bounded domain,
and
, by applying direct moving plane methods. Finally, we show the relations between the classical operator − Δ and the nonlocal operator in ((1)) as *α*2. Copyright © 2016 John Wiley & Sons, Ltd.

This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations

where *α* ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech-type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, *α* = 1,*β* = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we propose a space-time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space-time spectral-Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line *x* > as follows:

where *l* is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of *l* > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of *l* < 0. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a parallel fiber-reinforced periodic elastic composite that present an imperfect contact of spring type between the fiber and the matrix. We use the elliptic integral of Cauchy type for solving the plane strain local problems that arise from the asymptotic homogenization method. Several general conditions are assumed, which include that the fibers are disposed of arbitrary manner in the local cell, that all fibers present contact perfect with different constants of imperfection, and that their cross section is a smooth closed arbitrary curve. We find that there are infinity solutions for these problems, and we find relations between these solutions and effective coefficients of the composite. Finally, we obtain analytic formulae for the circular fiber case and show some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider a nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The fluid domain is a subset of **R**^{3} bounded with two coaxial cylinders that present solid thermoinsulated walls. The mathematical model is set up in Lagrangian description. If we assume that the initial mass density, temperature, as well as the velocity and microrotation vectors are smooth enough cylindrically symmetric functions, then our problem has a generalized cylindrically symmetric solution for a sufficiently small time interval. Here, we prove the uniqueness of this solution. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents all possible exact explicit peakon, pseudo-peakon, cuspon and smooth solitary wave solutions for a nonlocal Kerr-like media. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. We present peakon, pseudo-peakon, cuspon soliton solution in an explicit form. We also have obtained smooth soliton. Mathematical analysis and numeric graphs are provided for those soliton solutions of the nonlocal Kerr-like media. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we prove a global well posedness of the three-dimensional incompressible Navier–Stokes equation under an initial data, which belong to the non-homogeneous Fourier–Lei–Lin space
for *σ*⩾ − 1 and if the norm of the initial data in the Lei–Lin space
is controlled by the viscosity. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this work is to study *μ*-pseudo almost automorphic solutions of abstract fractional integro-differential neutral equations with an infinite delay. Thanks to some restricted hypothesis on the delayed data in the phase space, we ensure the existence of the ergodic component of the desired solution. Copyright © 2016 John Wiley & Sons, Ltd.

The equations describing the steady flow of Cosserat–Bingham fluids are considered, and existence of weak solution is proved for the three-dimensional boundary-value problem with the use of the Lipschitz truncation argument. In contrast to the classical Bingham fluid, the micropolar Bingham fluid supports local micro-rotations and two types of plug zones. Our approach is based on an approximation of the constitutive relation by a generalized Newtonian constitutive relation and a subsequent limiting process. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The global stability of equilibria is investigated for a nonlinear multi-group epidemic model with latency and relapses described by two distributed delays. The results show that the global dynamics are completely determined by the basic reproduction number under certain reasonable conditions on the nonlinear incidence rate. Moreover, compared with the results in Michael Y. Li and Zhisheng Shuai, Journal Differential Equations 248 (2010) 1–20, it is found that the two distributed delays have no impact on the global behaviour of the model. Our study improves and extends some known results in recent literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial-homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non-existence of traveling waves is obtained by two-sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non-isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial-boundary value problem of the non-isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero-order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.

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