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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]

Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [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]

Abstract In this paper, we show the existence of the first nontrivial family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in F. de la Hoz et al. (2016) [13]. [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 introduce a new concept of completely linear degeneracy for quasilinear hyperbolic systems in several space variables, and then get an interesting property for multidimensional hyperbolic conservation laws satisfying our new definition. For applications, we give some examples arising from mathematics and physics at last. [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 consider the following nonlinear coupled elliptic system ( A ε ) { − ε 2 Δ u + P ( x ) u = μ 1 u 3 + β u v 2 in R N , − ε 2 Δ v + Q ( x ) v = μ 2 v 3 + β u 2 v in R N , u > 0 , v > 0 in R N , u → 0 , v → 0 as | x | → + ∞ , where ε > 0 is a parameter, μ 1 , μ 2 > 0 and β > 0 are constants, and P ( x ) and Q ( x ) are two nonnegative, smooth functions with different nondegenerate critical points and separated zero sets. Due to the Lyapunov–Schmidt reduction method and the Maximum Principle, we show that when β is less than a small positive number, there exists an ε 0 > 0 such that for any 0 < ε < ε 0 , ( A ε ) has a segregated vector solution where ( u ε , v ε ) and u ε is trapped in a neighborhood of the nondegenerate critical points of P ( x ) as well as the zero sets of P ( x ) , and v ε is trapped in a neighborhood of the nondegenerate critical points of Q ( x ) as well as the zero sets of Q ( x ) . Moreover, the amplitudes of u ε (res v ε ) around the nondegenerate critical points and the zero sets of P ( x ) (res Q ( x ) ) are of a different order compared with ε . To the best of our knowledge, these multi-scale solutions to the system have not been obtained previously. [ABSTRACT FROM AUTHOR]

This paper presents the major achievements of the 20th century regarding Karamata functions and the theory of differential equations, made mostly by V. Marić, M. Tomić, E. Omey, J.L. Geluk. The connection between these notions was first noticed by V.G. Avakumović (1910–1990). Slowly and regularly varying functions were introduced by J. Karamata (1902–1967). A group of mathematicians from the Karamata School of classical mathematical analysis were pioneers in research on these functions and their role in the theory of differential equations. Special attentions is given to the study of the Thomas–Fermi, Emden–Fowler and Friedmann equations, as well as the classical second order linear differential equations. [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, we study the asymptotic behavior of solutions of a class of non-autonomous delay differential equations. These equations have important practical applications and generalize those on which Bernfeld and Haddock conjectured that each solution of the equations tends to a constant. It is shown that every solution of the equations is bounded and tends to a constant as t → + ∞ , which improves and extends some existing ones. [ABSTRACT FROM AUTHOR]

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 considers stochastic population dynamics driven by Lévy noise. The contributions of this paper lie in that: (a) Using the Khasminskii–Mao theorem, we show that the stochastic differential equation associated with our model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such a model. [Copyright &y& Elsevier]

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: The aim of this paper is to investigate the possibility of solving a linear differential equation of degree by means of differential equations of degree less than or equal to a fixed , . This paper recovers and extends work of G. Fano, M. F. Singer and E. Compoint. Representations of algebraic Lie algebras are the main tool. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the growth of two composite entire functions of finite iterated order and a series of comparative growths of () with and . At the end of this paper, we apply some growth results into the factorization of the solutions of linear differential equation. We achieve some results which are the improvements and extensions of the previous results. [Copyright &y& Elsevier]

Abstract: In the beginning of the 1990s we devoted a sequence of papers to perturbation theory, singular limits and well-posedness problems. In particular, the strong well-posedness of the initial–boundary value problem for the compressible Euler equations was demonstrate for the first time. Our method also allowed singular limit results in the strong norm, even under assumptions weaker than the current ones in the literature (where the strong norm is not reached). It is worth noting that, until now, the above method and results have not been substantially improved. Hence an introduction to it still looks timely. Actually, in a forthcoming paper, by returning to this method, we improve (in a very substantial way) some important results recently appeared in the literature. [Copyright &y& Elsevier]

Abstract: Some exact solutions to the first, second and extended Stefan problems with fractional time derivative described in the Caputo sense are given by means of fractional Green''s function and Wright function in this paper. By the aid of simple calculations, many results of differential equations of integer order can be obtained as special cases of the results given by this paper. [Copyright &y& Elsevier]

Abstract: We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28 (2005) 917–945] where an energy a priori estimate for the linearized free boundary value problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in [S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622–629 (in Russian); W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655]. In this paper we derive an a priori estimate in the whole stability domain. The crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss'' type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition, i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of derivatives in the energy estimate. The result of this paper is a necessary step to prove the local-in-time existence of stable nonplanar incompressible current-vortex sheets by a suitable Nash–Moser type iteration scheme. [Copyright &y& Elsevier]

Abstract: The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. This theorem was weakened recently by Argyros who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions of the Kantorovich theorem. In the present paper we prove a weak Kantorovich-type theorem that gives the same conclusions as the previous two results under weaker conditions. Illustrative examples are provided in the paper. [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: In this paper, we deal with the problem of uniqueness and weighted sharing of two meromorphic functions with their first derivatives having the same fixed points with the same multiplicities. The results in this paper improve those given by K. Tohge, Xiao-Min Li and Hong-Xun Yi. [Copyright &y& Elsevier]

Abstract: In this paper, by means of a series of counterexamples, we study in a systematic way the relationships among (pseudo, quasi) α-preinvexity, (strict, strong, pseudo, quasi) α-invexity and (strict, strong, pseudo, quasi) αη-monotonicity. Results obtained in this paper can be viewed as a refinement and improvement of the results of Noor and Noor [M.A. Noor, K.I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316 (2006) 697–706]. [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]

Abstract: In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation on a time scale ; here is a quotient of odd positive integers with and real-valued positive functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. [Copyright &y& Elsevier]

This paper considers a semilinear integro-differential equation of Volterra type which interpolates semilinear heat and wave equations. Global existence of solutions is showed in spaces of Besov type based in Morrey spaces, namely Besov–Morrey spaces. Our initial data is larger than the previous works and our results provide a maximal existence class for semilinear interpolated heat-wave equation. Some symmetries, self-similarity and asymptotic behavior of solutions are also investigated in the framework of Besov–Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the existence and uniqueness of L p ( p > 1 ) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in y for the generator g . Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of L p ( p > 1 ) solutions for this kind of BSDEs. [ABSTRACT FROM AUTHOR]

It is well-known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations. In the present work, a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem. The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove existence of optimal design for the new model. This work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional (see [14] ). [ABSTRACT FROM AUTHOR]

The Bohl exponents, similarly as the Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. By contrast with properties of the Lyapunov characteristics, properties of the Bohl exponents are much less investigated. In this paper we present some new properties of the upper Bohl exponents of the diagonal discrete time-varying linear system. [ABSTRACT FROM AUTHOR]

Tachiya investigated a class of infinite products of rational functions arithmetically and established that their values at certain algebraic points are algebraic numbers if and only if the infinite products are rational functions. In this paper we prove further arithmetical results for the values of these infinite products both qualitatively and quantitatively, which can be carried out by studying these infinite products as formal power series carefully. [ABSTRACT FROM AUTHOR]

Let f ( n ) be a multiplicative function satisfying | f ( n ) | ≤ 1 , q ( ≤ N 2 ) be a prime number and a be an integer with ( a , q ) = 1 , χ be a non-principal Dirichlet character modulo q . In this paper, we shall prove that ∑ n ≤ N f ( n ) χ ( n + a ) ≪ N q 1 4 log ⁡ log ⁡ ( 6 N ) + q 1 4 N 1 2 log ⁡ ( 6 N ) + N log ⁡ log ⁡ ( 6 N ) and that ∑ n ≤ N f ( n ) χ ( n + a 1 ) ⋯ χ ( n + a t ) ≪ N q 1 4 log ⁡ log ⁡ ( 6 N ) + q 1 4 N 1 2 log ⁡ ( 6 N ) + N log ⁡ log ⁡ ( 6 N ) , where t ≥ 2 , a 1 , … , a t are pairwise distinct integers modulo q . [ABSTRACT FROM AUTHOR]

The paper is concerned with orders of the Kolmogorov, Gelfand and linear widths of function classes on sets with tree-like structure. As examples we consider weighted Sobolev classes on a John domain and some function classes on trees. [ABSTRACT FROM AUTHOR]

In this paper we prove two global existence results for Hughes' model for pedestrian flows under assumptions that ensure that the traces of the solutions along the turning curve are zero for all positive times. [ABSTRACT FROM AUTHOR]

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]

• New analysis of the ligand-induced dimerisation process for VEGFR is presented. • A key idea in the mathematical model and analysis is kinetic binding cooperativity. • The nonlinear model is in contrast with the corresponding linear model for GPCRs. • Numerical and asymptotic analysis reveal insights into the underlying LID dynamics. • Our simulations show excellent fit to recent experimental data for VEGFR. The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and many other diseases. It is widely accepted that VEGFR receptors dimerise in response to VEGF binding. However, analysis of these mechanisms and their implications for drug development still requires further exploration. In this paper, we present a mathematical model representing the binding of VEGF to VEGFR and the subsequent ligand-induced dimerisation. A key factor in this work is the qualitative and quantitative effect of binding cooperativity, which describes the effect that the binding of a ligand to a receptor has on the binding of that ligand to a second receptor, and the dimerisation of these receptors. We analyse the ordinary differential equation system at equilibrium, giving analytical solutions for the total amount of ligand bound. For time-course dynamics, we use numerical methods to explore possible behaviours under various parameter regimes, while perturbation analysis is used to understand the intricacies of these behaviours. Our simulation results show an excellent fit to experimental data, towards validating the model. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation. By establishing a compactness lemma and using variational methods, the existence result of two homoclinic solutions is obtained under some suitable assumptions, but not requiring the periodicity condition. Some recent results are improved and extended. [Copyright &y& Elsevier]

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: The coexistence of heteroclinic orbits and saddle foci is concerned with the basic assumption in Shilʼnikov heteroclinic theorem. Two aspects of this discussion are conducted in the paper. Firstly, many third-order systems, which possess exact heteroclinic orbits expressed by pure hyperbolic functions or the combination of hyperbolic and triangle functions and so on, have been constructed. At the same time, the existence of saddle foci is tested and some problems are proposed. Secondly and more importantly, the existence of heteroclinic orbits to saddle foci is studied. The necessary condition for the coexistence of heteroclinic orbits and saddle foci is obtained. Finally, an example is given to show the effectiveness of the results, and some conclusions and problems are presented. [Copyright &y& Elsevier]

Abstract: Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section. [Copyright &y& Elsevier]

Abstract: In the present paper, we study the Modified Nonlinear Schrödinger Equations (MNSE). Without any growth condition on the nonlinear term, we obtain the existence of infinitely many small solutions for MNSE by a dual approach. [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]