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In this paper, we apply level set and nonlinear perturbation methods to obtain the asymptotic behavior of the solution to a kind of parabolic Monge–Ampère equation at infinity. The Jörgens–Calabi–Pogorelov theorem for parabolic and elliptic Monge–Ampère equation can be regarded as special cases of our result. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Martínez, Samà and Simó (2006) [15] and results of Hu, Long and Sun (2014) [6] can be applied to these solutions of the charged three-body problem to get their linear stability. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary conditions for Lie-stability of the equilibrium solutions of autonomous and periodic Hamiltonian systems with n degrees of freedom which possess multiple resonances of arbitrary order. Necessary conditions for instability in the sense of Lyapunov in some cases of multiple resonances are provided. These conditions depend on the order of the resonance and the coefficients of the Hamiltonian function. [ABSTRACT FROM AUTHOR]

This paper is concerned with the critical conditions of nonlinear elliptic equations and the corresponding integral equations involving Riesz potentials and Bessel potentials. We show that the equations and some energy functionals are invariant under the scaling transformation if and only if the critical conditions hold. In addition, the Pohozaev identity shows that those critical conditions are the necessary and sufficient conditions for existence of the finite energy positive solutions. Finally, we discuss respectively the existence of the negative solutions of the k -Hessian equations in the subcritical case, critical case and supercritical case. Here the Serrin type critical exponent and the Sobolev type critical exponent play key roles. [ABSTRACT FROM AUTHOR]

Consider planar real polynomial differential equations of the form x ˙ = L x + X n ( x ) , where x = ( x , y ) ∈ R 2 , L is a 2 × 2 matrix and X n is a homogeneous vector field of degree n > 1 . Most known results about these equations, valid for infinitely many n , deal with the case where the origin is a focus or a node and give either non-existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin. In this paper we improve some of these results and moreover we show that for n ≥ 3 odd there are equations of this form having at least ( n + 1 ) / 2 limit cycles surrounding the origin. Our results include cases where the origin is a focus, a node, a saddle or a nilpotent singularity. We also discuss a mechanism for the bifurcation of limit cycles from infinity. [ABSTRACT FROM AUTHOR]

There is a wide literature involving the study of isochronous equations of the type &xuml;(t) + V′(x(t)) = 0, where V is a C² -function. In this paper we show how the kinetic energy T(y) = 1/2y² can be modified still preserving the isochronicity property of the corresponding system. More generally we provide estimates for the periods, and show an application to the Steen's equation and other systems related to the anharmonic potential V(x)=ax²+bx−2. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial–boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H 2 initial data. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized quasilinear Choquard equation − ε p Δ p u + V ( x ) | u | p − 2 u = ε μ − N ( ∫ R N Q ( y ) F ( u ( y ) ) | x − y | μ ) Q ( x ) f ( u ) in R N , where Δ p is the p-Laplacian operator, 1 < p < N , V and Q are two continuous real functions on R N , 0 < μ < N , F ( s ) is the primate function of f ( s ) and ε is a positive parameter. Under suitable assumptions on p , μ and f , we establish a new concentration behavior of solutions for the quasilinear Choquard equation by variational methods. The results are also new for the semilinear case p = 2 . [ABSTRACT FROM AUTHOR]

In this paper we provide a characterization of local integrability for analytic or formal differential systems in R n or C n via the integrability varieties. Our result generalizes the classical one of Poincaré and Lyapunov on local integrability of planar analytic differential systems to any finitely dimensional analytic differential systems. As an application of our theory we study the integrability of a family of four-dimensional quadratic Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We consider a reaction–diffusion–advection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some non-degeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is one-dimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this non-standard system is obtained when one of the diffusion rates is large. [ABSTRACT FROM AUTHOR]

This paper deals with the spectral properties of the Sturm–Liouville operators generated by the differential expression L ( y ) = 1 w ( − ( p ( x ) y ′ ) ′ + υ ( x ) y ) with singular coefficients υ ( x ) in the sense of distributions. In particular, we study the operators with δ -type point interactions at the centers x k on the positive half line in terms of energy forms. Necessary and sufficient conditions and also simple sufficient conditions are given for the spectrum of the operators to be discrete. We also prove sufficient conditions on the stability of the continuous spectrum. [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]

Abstract: We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal forms of CDEs under topological equivalence. Generic CDEs are important in the study of slow–fast (SF) systems. Many properties and the characteristic behavior of the solutions of SF systems can be inferred from the corresponding CDE. Therefore, the results of this paper show a first approximation of the flow of generic SF systems with three slow variables. [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: 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: Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy. [Copyright &y& Elsevier]

Abstract: In this paper we study the quasi-homogeneous polynomial differential systems and provide an algorithm for obtaining all these systems with a given degree. Using this algorithm we obtain all quasi-homogeneous vector fields of degree 2 and 3. The quasi-homogeneous polynomial differential systems are Liouvillian integrable. In particular, we characterize all the quasi-homogeneous vector fields of degree 2 and 3 having a polynomial, rational or global analytical first integral. [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 consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. The autonomous dynamical system is described by the following infinite system of second order differential equations where denotes the interaction potential between two neighboring particles and is the state of the i-th particle. Supposing is superquadratic at infinity, for all , we obtain a nonzero T-periodic solution of finite energy which may be nonconstant in some range of period. If in addition is even in x, we also obtain infinitely many geometrically distinct solutions for any period . In particular, a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period. Since the functional associated to the above system is invariant under the actions of the non-compact group and the continuous compact group under our assumptions, in order to prove our results, we need to extend the abstract critical point theorem about strongly indefinite functional developed by Bartsch and Ding [Math. Nachr. 279 (2006) 1267–1288] to a more general class of symmetry. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the investigation of the dynamics of non-autonomous differential equations. The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous – and autonomous – differential equations (one for each global solution in the attractor of the driving semigroup of the associated skew product semiflow). In some cases, we have infinitely many limiting problems (in contrast with the autonomous – or asymptotically autonomous – case for which we have only one limiting problem; that is, the semigroup itself). We concentrate our attention in the study of the Morse decomposition of attractors for these non-autonomous limiting problems as a mean to understand some of the asymptotics of our non-autonomous differential equations. In particular, we derive a Morse decomposition for the global attractors of skew product semiflows (and thus for pullback attractors of non-autonomous differential equations) from a Morse decomposition of the attractor for the associated driving semigroup. Our theory is well suited to describe the asymptotic dynamics of non-autonomous differential equations defined on the whole line or just for positive times, or for differential equations driven by a general semigroup. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces , , (which generalize the Sobolev spaces ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with , , ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to derive a wall law for a flow over a very rough surface. We consider a viscous incompressible fluid filling a 3-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a very rough wall. The latter consists in a plane wall covered with periodically distributed asperities which size depends on a small parameter and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using asymptotic expansions and boundary layer correctors we construct and analyze an asymptotic approximation of order ( being arbitrary small) in the -norm for the velocity and in the -norm for the pressure. We derive an effective boundary condition of Navier type, then expressing the boundary layer terms in terms of the homogenized solution and the solution of a cell problem we obtain an effective approximation in the whole domain of the flow. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the scattering problem for the higher order Schrödinger operator , where is a long-range potential. At this moment the classical wave operators do not exist, so we provide one kind of modified wave operators and show the existence and the asymptotic completeness of such wave operators, as well as the expression of the corresponding scattering operator. [Copyright &y& Elsevier]

Abstract: In this paper, we study the stability property of Hamiltonian systems on the Wasserstein space. Let H be a given Hamiltonian satisfying certain properties. We regularize H using the Moreau–Yosida approximation and denote it by . We show that solutions of the Hamiltonian system for converge to a solution of the Hamiltonian system for H as τ converges to zero. We provide sufficient conditions on H to carry out this process. [Copyright &y& Elsevier]

Abstract: This paper studies the global regularity of classical solutions to the 2D incompressible magnetohydrodynamic (MHD) equations with horizontal dissipation and horizontal magnetic diffusion. It is shown here that the horizontal component of any solution admits a global (in time) bound in any Lebesgue space with and the bound grows no faster than the order of as r increases. In addition, we establish a conditional global regularity in terms of the -norm of the horizontal component and the global regularity of a slightly regularized version of the aforementioned MHD equations. [Copyright &y& Elsevier]

Abstract: The Harmonic Balance Method provides a heuristic approach for finding truncated Fourier series as an approximation to the periodic solutions of ordinary differential equations. Another natural way for obtaining these types of approximations consists in applying numerical methods. In this paper we recover the pioneering results of Stokes and Urabe that provide a theoretical basis for proving that near these truncated series, whatever is the way they have been obtained, there are actual periodic solutions of the equation. We will restrict our attention to one-dimensional non-autonomous ordinary differential equations, and we apply the obtained results to a concrete example coming from a rigid cubic system. [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 we show the existence of a positive solution for the following class of elliptic equations where f is a continuous function with quasicritical growth and K is a nonnegative function verifying some conditions of two kinds. We analyze the problem with K asymptotically periodic, K periodic and K belonging to some . [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: We consider the p-system of isentropic gas dynamics. One of the outstanding questions in the study of one-dimensional Euler equations is the BV-existence and local structure of solutions having large data, including the vacuum state. The author has recently given a full description of pairwise wave interactions in gas dynamics, which includes uniform interaction estimates up to vacuum. In this paper we consider composite interactions, which can be regarded as a degenerate superposition of pairwise interactions. We construct a class of weak solutions which demonstrate some interesting and surprising features, such as a shock of one family disappearing and a shock of the opposite family emerging. We give precise quantitative conditions which determine the outgoing waves. We also construct weak solutions of the p-system which demonstrate the collapse of a vacuum: in most cases two shocks will emerge from the vacuum, but in certain asymmetric cases a single shock and a rarefaction may emerge. We emphasize that the solutions constructed here are both explicit and exact weak solutions to the Euler equations of isentropic gas dynamics. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the large time behavior and especially to the regularity of the global attractor of the second grade fluid equations in the two-dimensional torus. We first recall that, for any size of the material coefficient , these equations are globally well posed and admit a compact global attractor in . We prove that, for any , there exists , such that belongs to if the forcing term is in . We also show that this attractor is contained in any Sobolev space provided that α is small enough and the forcing term is regular enough. These arguments lead also to a new proof of the existence of the compact global attractor . Furthermore we prove that on , the second grade fluid system can be reduced to a finite-dimensional system of ordinary differential equations with an infinite delay. Moreover, the existence of a finite number of determining modes for the equations of the second grade fluid is established. [Copyright &y& Elsevier]

Abstract: Recurrent nonautonomous two-dimensional systems of differential equations of ordinary, finite delay and reaction–diffusion types given by cooperative and concave vector fields define monotone and concave skew-product semiflows, whose dynamics is analyzed in this paper. A complete description of all the minimal sets in a very interesting dynamical situation is provided, and some criteria to determine its possible existence and the area on which they lie are given. Suitable examples prove the optimality of the results. [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: We consider the Hardy–Hénon equation with and and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space . It has been conjectured that this property is true if (and only if) , where is the Hardy–Sobolev exponent, given by . However, when , the conjecture had up to now been proved only for . Indeed the case seems more difficult, due to . In this paper, we prove the conjecture for in dimension , in the case of bounded solutions. Next, for the conjecture in the case , and for related estimates near isolated singularities and at infinity, we give new proofs – based in particular on doubling-rescaling arguments – and we provide some extensions of these estimates. These proofs are significantly simpler than the previously known ones. Finally, we clarify some of the previous results on a priori estimates for the related Dirichlet problem. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a Dancer-type unilateral global bifurcation result for one-dimensional p-Laplacian problem Under some natural hypotheses on the perturbation function , we show that is a bifurcation point of the above problem and there are two distinct unbounded continua, and , consisting of the bifurcation branch from (), where is the k-th eigenvalue of the linear problem corresponding to the above problem. As the applications of the above result, we study the existence of nodal solutions for the following problem Moreover, based on the bifurcation result of Girg and Takáč (2008) , we prove that there exist at least a positive solution and a negative one for the following problem [Copyright &y& Elsevier]

Abstract: In this paper, we consider two classes of free boundary value problems of a viscous two-phase liquid-gas model relevant to the flow in wells and pipelines with mass-dependent viscosity coefficient. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. We obtain the asymptotic behavior and decay rates of the mass functions , when the initial masses are assumed to be connected to vacuum both discontinuously and continuously, which improves the corresponding result about Navier–Stokes equations in Zhu (2010) . [Copyright &y& Elsevier]

Abstract: In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results. [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 mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincaréʼs one (Poincaré, 1891, 1897 ), the Furtaʼs one (Furta, 1996 ), part of Chen et al.ʼs ones (Chen et al., 2008 ), and the Shiʼs one (Shi, 2007 ). The key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation. [Copyright &y& Elsevier]

Abstract: For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401–1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have? In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.ʼs as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem. [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: This paper is concerned with the large time behavior of solutions to a radiating gas model, which is represented mathematically as a Cauchy problem for a one-dimensional hyperbolic–elliptic coupled system, with suitably given far field states. Suppose the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, then we can construct a “viscous contact wave” for such a hyperbolic–elliptic system. Based on the energy methods and the ellipticity of the radiation flux equation, we prove that the “viscous contact wave” is asymptotically stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small. [Copyright &y& Elsevier]