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We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L 1 norm of potentials is known. We establish the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study a spatially heterogeneous predator–prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14] . We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m . Finally, we further compare the difference of steady-state solutions between m > 0 and m = 0 . It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if the solution of the planar N + 1 -body problem has N bodies rotating at the vertices of a regular N -gon then the N bodies have equal mass and the remaining body must be at the center. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ground state solution for an indefinite variational problem of the type ( P ) { − Δ u + ( V ( x ) − W ( x ) ) u = f ( x , u ) in R N , u ∈ H 1 ( R N ) , where N ≥ 2 , V , W : R N → R and f : R N × R → R are continuous functions verifying some technical conditions and f possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, V is Z N -periodic, W goes to 0 at infinity and f is asymptotically periodic. [ABSTRACT FROM AUTHOR]

In this paper, we consider the radial singular solution of Δ 2 u = − 1 u p , in R N with u > 0 , p > 0 . By using some elementary ordinary differential equation arguments, we prove the above equation admits no radial singular solution for N = 3 , p ≥ 3 . In addition, the exact asymptotic behaviors of the radial singular solution as r → 0 is established for p = 1 , N ≥ 4 , which have a significant difference with the explicit singular solution. As a product, the singularity of any radial singular solution is of type II if p = 1 , N ≥ 4 . [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an integrable two-component peakon system, which was proposed by Xia, Qiao and Zhou. We present a precise blow-up scenario and a new blow-up result for strong solutions to the system. Moreover, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the quasilinear fully parabolic Keller–Segel system under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary, . The diffusivity is assumed to satisfy some further technical conditions such as algebraic growth and , which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that for with , and . When , this paper represents an improvement of Tao and Winkler [17], because the domain does not necessarily need to be convex in this paper. In the case and , uniform-in-time boundedness is an open problem left in a previous paper [7]. This paper also gives an answer to it in bounded domains. [Copyright &y& Elsevier]

In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ -versions as well as hyperelastic rod equation. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the existence and stability of periodic pyramidal traveling fronts for reaction–diffusion equations with bistable time-periodic nonlinearity in with . It is well known that two-dimensional periodic traveling curved fronts exist and are stable. In this paper, by constructing various of supersolutions and subsolutions, we first show that there exist three-dimensional periodic pyramidal traveling fronts, and then we prove that such periodic pyramidal traveling fronts are asymptotically stable. Finally, we further prove that our existence result holds for with . [Copyright &y& Elsevier]

Abstract: In this paper, we consider the following modified quasi-geostrophic equations where is a fixed parameter. This equation was recently introduced by P. Constantin, G. Iyer and J. Wu (2001) in as a modification of the classical quasi-geostrophic equation. In this paper, we prove that for any initial data in the Sobolev space , Eq. (MQG) has a global and smooth solution θ in . [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for , while Guo and Wu (1995) only considered the case . We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen–Lee–Lin equation and Gerdjikov–Ivanov equation, respectively. [Copyright &y& Elsevier]

Abstract: To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties. [Copyright &y& Elsevier]

Abstract: The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state. To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold. [Copyright &y& Elsevier]

We point out a few mistakes in our earlier paper [2] and propose some corrections. Our main results remain valid provided some more restrictive assumptions. [ABSTRACT FROM AUTHOR]

In this paper we consider analytic planar differential systems having a first integral of the form H(x,y)=A(x)+B(x)y+C(x)y2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of . We also give a refined blowup criterion in the supercritical case. [Copyright &y& Elsevier]

Abstract: We consider parameterized families of flows in locally compact metrizable spaces and give a characterization of those parameterized families of flows for which uniform persistence continues. On the other hand, we study the generalized Poincaré–Andronov–Hopf bifurcations of parameterized families of flows at boundary points of or, more generally, of an n-dimensional manifold, and show that this kind of bifurcations produce a whole family of attractors evolving from the bifurcation point and having interesting topological properties. In particular, in some cases the bifurcation transforms a system with extreme non-permanence properties into a uniformly persistent one. We study in the paper when this phenomenon happens and provide an example constructed by combining a Holling-type interaction with a pitchfork bifurcation. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional Laplacian where ( ) is a bounded smooth domain, , stands for the fractional Laplacian. When f is superlinear and subcritical, we prove the existence of a positive solution, a negative solution and a nodal solution. If is odd in u, we obtain an unbounded sequence of nodal solutions. In addition, the number of nodal domains of the nodal solutions are investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation where , and is an unbounded domain of , . We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence, which is used to obtain sharp results. [Copyright &y& Elsevier]

Abstract: In this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied. This family is an extension to higher degree of Loudʼs systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. We study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center. Moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period, giving also its character (maximum or minimum). Finally we propose a complete conjectural bifurcation diagram of the period function of these generalized Loudʼs centers. [Copyright &y& Elsevier]

Abstract: In this paper, we study the necessary conditions and sufficient conditions for the twisted angles of the central configurations formed by two twisted regular polygons, in particular, we prove that for the 2N-body problem, the twisted angles must be or . [Copyright &y& Elsevier]

Abstract: In this paper, the -index is defined and the -boundary value problem of Hamiltonian system is studied. As applications, the existence and multiplicity results of periodic solutions of asymptotically linear delay differential systems and delay Hamiltonian systems are obtained. [Copyright &y& Elsevier]

Abstract: In this paper isochronicity of centers is discussed for a class of discontinuous differential system, simply called switching system. We give some sufficient conditions for the system to have a regular isochronous center at the origin and, on the other hand, construct a switching system with an irregular isochronous center at the origin. We give a computation method for periods of periodic orbits near the center and use the method to discuss a switching Bautin system for center conditions and isochronous center conditions. We further find all of those systems which have an irregular isochronous center. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions on the initial data to guarantee blow-up phenomenon for the modified two-component Camassa–Holm (MCH2) system. [Copyright &y& Elsevier]

Abstract: In this paper, we use the potential analysis to study the properties of the positive solutions of a γ-Laplace system in Here , satisfy the critical condition . First, the positive solutions u and v satisfy an integral system involving the Wolff potentials. We then use the method of regularity lifting to obtain an optimal integrability for this Wolff type integral system. Different from the case of , it is more difficult to handle the asymptotics since u and v have not radial structures. We overcome this difficulty by a new method and obtain the decay rates of u and v as . We believe that this new method is appropriate to deal with the asymptotics of other decaying solutions without the radial structures. [Copyright &y& Elsevier]

Abstract: In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented. [Copyright &y& Elsevier]

Abstract: In this paper we study the maximal number of limit cycles in Hopf bifurcations for two types of Liénard systems and obtain an upper bound of the number. In some cases the upper bound is the least, called the Hopf cyclicity. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of infinitely many periodic solutions for second order Hamiltonian systems , where is either asymptotically quadratic or superquadratic as . [Copyright &y& Elsevier]

Abstract: The paper deals with solutions of a differential inclusion constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure co F satisfies a strict inward pointing condition at every boundary point . Given a reference trajectory taking values in an ε-neighborhood of Ω, we prove the existence of a second trajectory which satisfies . As shown by an earlier counterexample, this bound is sharp. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the initial boundary value problems of the system modeling the compressible viscoelastic fluid of Oldroyd type. The global in time solution is proved to exist uniquely near the equilibrium state in . [Copyright &y& Elsevier]

Abstract: Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero. In this paper we show that, for analytic systems, this result is of topological nature. This means that it is enough to impose a change of sign at the zero, without any assumption on the succesive derivatives. [Copyright &y& Elsevier]

Abstract: The paper studies the longtime behavior of the Kirchhoff type equation with a strong dissipation . It proves that the related continuous semigroup possesses in the phase space with low regularity a global attractor which is connected. And an example is shown. [ABSTRACT FROM AUTHOR]

Abstract: Consider a Navier–Stokes liquid filling the three-dimensional space exterior to a moving rigid body and subject to an external force. Using a coordinates system attached to the body, the equations of the fluid can be written in a time-independent domain, which results in a perturbed Navier–Stokes system where the extra terms depend on the velocity of the rigid body. In this paper, we consider the related whole space problem and construct a strong solution with finite kinetic energy for the corresponding steady-state equations. For this, appropriate conditions on the external force have to be imposed (for instance, that it is a function with compact support and null average) together with a smallness condition involving the viscosity of the fluid. First, a linearized version of the problem is analysed by means of the Fourier transform, and then a strong solution to the full nonlinear problem is obtained by a fixed point procedure. We also show that such a solution satisfies the energy equation and is unique within a certain class. [Copyright &y& Elsevier]

Abstract: We consider a mechanical system which is controlled by means of moving constraints. Namely, we assume that some of the coordinates can be directly assigned as functions of time by implementing frictionless constraints. This leads to a system of ODE''s whose right hand side depends quadratically on the time derivative of the control. In this paper we introduce a simplified dynamics, described by a differential inclusion. We prove that every trajectory of the differential inclusion can be uniformly approximated by a trajectory of the original system, on a sufficiently large time interval, starting at rest. Under a somewhat stronger assumption, we show this second trajectory reaches exactly the same terminal point. [Copyright &y& Elsevier]

Abstract: The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions. [Copyright &y& Elsevier]

Abstract: This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where is a prime p or a power of a prime , and show that these systems can have a focus order ; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too. [Copyright &y& Elsevier]

Abstract: This paper deals with existence and exponential decay of homoclinic orbits in the first-order Hamiltonian system where the Hamiltonian function is nonperiodic in and superquadratic in . With certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems. [Copyright &y& Elsevier]

Abstract: This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front. [Copyright &y& Elsevier]

Abstract: This paper is contributed to the elliptic equation where , , , and is a constant. We study the structure of positive radial solutions of (0.1) and obtain the uniqueness of solution decaying faster than at ∞ if μ is small enough under some assumptions on K and f, where m is the slow decay rate. [Copyright &y& Elsevier]

Abstract: In this paper we study a class of Caffarelli–Kohn–Nirenberg inequalities without restricting the pertinent parameters. In particular, we determine the values of the corresponding optimal constants and the functions that achieve them, i.e., minimizers of a suitable functional. By studying a corresponding Euler–Lagrange equation, we also determine infinitely many sign-changing solutions at higher energy levels in addition to the found ground-state solutions. [Copyright &y& Elsevier]

Abstract: Under some nondegeneracy conditions we give asymptotic formulae for the stability parameter of a family of singular-limit Hill''s equation which depends on three parameters. We use the blow-up techniques introduced in [R. Martínez, A. Samà, C. Simó, Analysis of the stability of a family of singular-limit linear periodic systems in . Applications, J. Differential Equations 226 (2006) 652–686]. The main contribution of this paper concerns the study of the nondegeneracy conditions. We give a geometrical interpretation of them, in terms of heteroclinic orbits for some related systems. In this way one can determine values of the parameters such that the nondegeneracy conditions are satisfied. As a motivation and application we consider the vertical stability of homographic solutions in the three-body problem. [Copyright &y& Elsevier]

Abstract: In this paper we focus on the initial–boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial–boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role. [Copyright &y& Elsevier]

Abstract: Consider in this paper a linear skew-product system where or , and is a topological dynamical system on a compact metrizable space W, and where satisfies the cocycle condition based on θ and is continuously differentiable in t if . We show that ‘semi λ-exponential dichotomy’ of implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ-stable direction and if is constant a.e., then Θ is almost λ-exponentially dichotomous. To prove this, we first use Liao''s spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual -cocycles based on a uniquely ergodic endomorphism, and of -cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of -topology we obtain the density of -cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition. [Copyright &y& Elsevier]

Abstract: We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185–242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder. [Copyright &y& Elsevier]

Abstract: In this paper we consider the stationary Navier–Stokes equations in a bounded domain with a variable viscosity. We prove that one can uniquely determine the viscosity function from the knowledge of boundary data. [Copyright &y& Elsevier]

Abstract: In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles. [Copyright &y& Elsevier]

Abstract: In this paper a half space problem for the one-dimensional Boltzmann equation with specular reflective boundary condition is investigated. It is shown that the solution of the Boltzmann equation time-asymptotically converges to a global Maxwellian under some initial conditions. Furthermore, a time-decay rate is also obtained. [Copyright &y& Elsevier]