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We consider a C ∞ family of planar vector fields { X μ ˆ } μ ˆ ∈ W ˆ having a hyperbolic saddle and we study the Dulac map D (s ; μ ˆ) and the Dulac time T (s ; μ ˆ) from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio λ of the saddle plays an important role, we treat it as an independent parameter, so that μ ˆ = (λ , μ) ∈ W ˆ = (0 , + ∞) × W , where W is an open subset of R N. For each μ ˆ 0 ∈ W ˆ and L > 0 , the functions D (s ; μ ˆ) and T (s ; μ ˆ) have an asymptotic expansion at s = 0 and μ ˆ ≈ μ ˆ 0 with the remainder being uniformly L -flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on μ ˆ that can be shown to be C ∞ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter μ ˆ 0. Each coefficient has its own domain and it is of the form ((0 , + ∞) ∖ D) × W , where D a discrete set of rational numbers at which a resonance of the hyperbolicity ratio λ occurs. In our main result, Theorem A, we provide explicit expressions for some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at D × W and we give the corresponding residue, that plays an important role when compensators appear in the principal part. Furthermore we prove a result, Corollary B, showing that in the analytic setting each coefficient given in Theorem A is meromorphic on (0 , + ∞) × W and has only poles, of order at most two, along D × W. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is considered in a complex domain, and the asymptotic behavior of solutions with respect to the parameter is studied. Under suitable conditions, a formal power series solution is constructed, and then it is proved that this formal solution is summable in a suitable direction. This guarantees the existence of a holomorphic solution that admits the formal power series solution as an asymptotic expansion of Gevrey type. It is then proved that this solution also has a Gevrey type asymptotic expansion with respect to the parameter. [ABSTRACT FROM AUTHOR]

This paper deals with the period function of the reversible quadratic centers X ν = − y (1 − x) ∂ x + (x + D x 2 + F y 2) ∂ y , where ν = (D , F) ∈ R 2. Compactifying the vector field to S 2 , the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Π ν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of X ν that tend to Π ν 0 in the Hausdorff sense as ν → ν 0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν = (D , F) ∈ R 2 outside the segments { − 1 } × [ 0 , 1 ] and { 0 } × [ 0 , 2 ]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν ∈ R 2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u ″ (t) + A u (t) + b (t) u ′ (t) = 0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b (t) ∼ (1 + t) − β (− 1 < β < 1). In the case b (t) ≡ 1 , it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v ′ (t) + A v (t) = F n (t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b (t) v ′ (t) + A v (t) = 0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n -th order profile u ˜ n is given via v ′ (t) + m n (t) A v (t) = F n (t) which coefficient m n (time-scale) differs each other. [ABSTRACT FROM AUTHOR]

In this paper, we study heteroclinic bifurcation of limit cycles in a planar cubic near-Hamiltonian system by higher-order Melnikov functions. We compute the asymptotic expansion of the third-order Melnikov function near the heteroclinic loop L s and prove that this system can have five limit cycles around L s with proper perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via the Agrachev–Sarychev method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the defocusing Schrödinger (NLS) equation with a nonzero background i q t + q x x − 2 (| q | 2 − 1) q = 0 , q (x , 0) = q 0 (x) , lim x → ± ∞ ⁡ q 0 (x) = ± 1. Recently, for the space-time region | x / (2 t) | < 1 which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the N -soliton solutions for the NLS equation by using the ∂ ¯ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form (0.1) q (x , t) = T (∞) − 2 q s o l , N (x , t) + O (t − 1) , whose leading term is N-soliton and the second term O (t − 1) is a residual error from a ∂ ‾ -equation. In this paper, we are interested in the large-time asymptotics in the space-time region | x / (2 t) | > 1 which is outside the soliton region, but there will be two stationary points appearing on the jump contour R. We found an asymptotic expansion that is different from (0.1) (0.2) q (x , t) = e − i α (∞) (1 + t − 1 / 2 h (x , t)) + O (t − 3 / 4) , whose leading term is a nonzero background, the second t − 1 / 2 order term is from the continuous spectrum and the third term O (t − 3 / 4) is a residual error from a ∂ ‾ -equation. The above two asymptotic results (0.1) and (0.2) imply that the region | x / (2 t) | < 1 considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region | x / (2 t) | > 1 considered by us is a slow decaying nonzero background region. • We obtain large-time asymptotics in the solitonless region |x/(2t)| > 1 for defocusing NLS equation with nonzero background. • This result is a complete supplement to the result recently obtained by Cuccagna and Jenkins. • | x / (2 t) | < 1 considered by Cuccagna and Jenkins is a fast decaying soliton region, while ours | x / (2 t) | > 1 is a slow decaying nonzero background region. [ABSTRACT FROM AUTHOR]

Let L u = u ″ + 2 A u ′ + B u be an ordinary differential operator with A and B are polynomials of any degree. In this paper we show that L is globally regular , i.e. L u ∈ S implies u ∈ S , for all tempered distribution u ∈ S ′ , if and only if the complex roots ξ ± (x) = i A (x) ± B (x) − A ′ (x) − (A (x)) 2 of the Weyl symbol σ L (x , ξ) = − ξ 2 + 2 i A (x) ξ + B (x) − A ′ (x) satisfy the condition lim | x | → + ∞ ⁡ | x Im ξ ± (x) | = + ∞. [ABSTRACT FROM AUTHOR]

Our aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one will be perturbed linearly. The required multiplicity can be achieved via folding of the system through certain choices of the periodicity of the coefficient matrix. Secondly, we apply such an asymptotic theory for the analysis of Floquet metamaterials. We provide a characterization of asymptotic exceptional points for a pair of subwavelength resonators with time-dependent material parameters. We prove that asymptotic exceptional points are obtained if the frequency components of the perturbations fulfill a certain ratio, which is determined by the geometry of the dimer of subwavelength resonators. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our preceding work on hyperbolic relaxation systems with characteristic boundaries of type I. Here we focus on the characteristic boundaries of type II, where the boundary is characteristic for the equilibrium system and is non-characteristic for the relaxation system. For this kind of characteristic initial-boundary-value problems (IBVPs), we introduce a three-scale asymptotic expansion to analyze the boundary-layer behaviors of the general multi-dimensional linear relaxation systems. Moreover, we derive the reduced boundary condition under the Generalized Kreiss Condition by resorting to some subtle matrix transformations and the perturbation theory of linear operators. The reduced boundary condition is proved to satisfy the Uniform Kreiss Condition for characteristic IBVPs. Its validity is shown through an error estimate involving the Fourier-Laplace transformation and an energy method based on the structural stability condition. [ABSTRACT FROM AUTHOR]

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles. [ABSTRACT FROM AUTHOR]

In this paper we obtain higher order asymptotic expansions of solutions to the Cauchy problem of the linear damped wave equation in R n u t t − Δ u + u t = 0 , u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x) , where n ∈ N and u 0 , u 1 ∈ L 1 (R n) ∩ L 2 (R n). Established hyperbolic effects seem to be new in the sense that the order of obtained expansions depends on the spatial dimension. [ABSTRACT FROM AUTHOR]

Abstract The Fourier transform, F , on R N (N ≥ 3) transforms the Cauchy problem for the strongly damped wave equation u t t − Δ u t − Δ u = 0 to an ordinary differential equation in time. We let u (t , x) be the solution of the problem given by the Fourier transform, and ν (t , ξ) be the asymptotic profile of F (u) (t , ξ) = u ˆ (t , ξ) found by Ikehata in the paper Asymptotic profiles for wave equations with strong damping (2014). In this paper we study the asymptotic expansions of the squared L 2 -norms of u (t , x) , u ˆ (t , ξ) − ν (t , ξ) , and ν (t , ξ) as t → ∞. With suitable initial data u (0 , x) and u t (0 , x) , we establish the rate of decay of the squared L 2 -norms of u (t , x) and ν (t , ξ) as t → ∞. By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence between u ˆ (t , ξ) and ν (t , ξ) in the L 2 -norm occurs quickly relative to their individual behaviors. This observation is similar to the diffusion phenomenon, which has been well studied. [ABSTRACT FROM AUTHOR]

Abstract The present paper is concerned with the asymptotic behaviors of radially symmetric solutions for the multi-dimensional Burgers equation on the exterior domain in R n , n ≥ 3 , where the boundary and far field conditions are prescribed. We show that in some case where the corresponding 1-D Riemann problem for the non-viscous part admits a shock wave, the solution tends toward a linear superposition of stationary and rarefaction waves as time goes to infinity, and also show the decay rate estimates. Furthermore, we improve the results on the asymptotic stability of the stationary waves which are treated in the previous papers [2] , [3]. Finally, for the case of n = 3 , we give the complete classification of the asymptotic behaviors, which includes even a linear superposition of stationary and viscous shock waves. [ABSTRACT FROM AUTHOR]

In this paper, we apply ∂ ‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equation u x t = u + 1 6 (u 3) x x , u (x , 0) = u 0 (x) ∈ H (R) , where H (R) = W 3 , 1 (R) ∩ H 2 , 2 (R) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale (y , t). In any fixed space-time cone C (y 1 , y 2 , v 1 , v 2) = { (y , t) ∈ R 2 : y = y 0 + v t , y 0 ∈ [ y 1 , y 2 ] , v ∈ [ v 1 , v 2 ] } , we compute the long time asymptotic expansion of the solution u (x , t) , which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N (I) -soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second | t | − 1 / 2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O (| t | − 1) from a ∂ ‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle. We give a structure theorem for the asymptotic expansion of the local Dulac time (as well as the local Dulac map) with the remainder uniformly flat with respect to the unfolding parameters. Here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable normal form can be used. The principal part of the asymptotic expansion is given in a monomial scale containing a deformation of the logarithm, the so-called Roussarie-Ecalle compensator. Especial attention is paid to the remainder's properties concerning the derivation with respect to the unfolding parameters. [ABSTRACT FROM AUTHOR]

We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of the rescaled Debye length, whose detailed derivation and analysis are carried out in our companion paper [8]. By establishing H m -norm, (m ≥ 2) , estimate of the difference between the original and approximation solutions, provided that the well-prepared initial data is given, we show that the local-in-time solution exists in the time interval, uniform in the quasi-neutral limit, and we prove the difference converges to zero with a certain convergence rate validating the formal expansion order. Our results mathematically justify the quasi-neutrality of a plasma in the regime of plasma sheath, indicating that a plasma is electrically neutral in bulk, whereas the neutrality may break down in a scale of the Debye length. [ABSTRACT FROM AUTHOR]

It is well-known that the 3D incompressible Euler equations admit some local-in-time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time-dependent hypersurface which evolves with the flow. In this paper we prove that such a solution can be obtained as zero-viscosity limit of strong solutions to the Navier-Stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit. Indeed we exhibit some sequences of exact solutions to the Navier-Stokes equations with vanishing viscosity which are given by multi-scale asymptotic expansions involving some characteristic boundary layers given by some linear PDEs. The convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the Euler equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive an original formula for the subwavelength resonance frequency of an encapsulated bubble with arbitrary shape in two dimensions. Using Gohberg-Sigal theory, we derive an asymptotic formula for this resonance frequency, as a perturbation away from the resonance of the uncoated bubble, in terms of the thickness of the coating. The formula is numerically verified in the case of circular bubbles, where the resonance can be efficiently computed using the multipole method. The approach involves the use of a pole-pencil decomposition of the leading order term in the asymptotic expansion of some integral operator in terms of the thickness of the coating, followed by the application of the generalized argument principle to find the characteristic value. This approach is quite general and can be applied to other subwavelength resonators such as coated plasmonic nanoparticles. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a previous paper by the authors. We are interested in the asymptotic behavior of eigenpairs on one dimensional linearized eigenvalue problem for Allen–Cahn equations as the diffusion coefficient tends to zero. We obtain the asymptotic profiles of all eigenfunctions by using the asymptotic formulas of corresponding eigenvalues, which have been obtained in the previous paper. Our results lead us to the concept of the classification of limiting eigenfunctions. In the case of Allen–Cahn equation it is provided by three special eigenfunctions, which correspond to the solutions of rescaled spectral problems on the whole line. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide asymptotic expansions of various spectral quantities of Zakharov–Shabat operators on the circle with new error estimates. These estimates hold uniformly on bounded subsets of potentials in Sobolev spaces. [ABSTRACT FROM AUTHOR]

The object of this work is to investigate the fluid mechanics of oblique impinging jet flows and to this end some existence and nonexistence results are initiated. First, we established the existence of incompressible oblique impinging jet plane flows with two asymptotic directions. More precisely, given a two-dimensional semi-infinitely long nozzle and a wall behind the nozzle, for any given mass flux in the inlet of the nozzle, then there exists a smooth incompressible oblique impinging jet flow with two asymptotic directions. The impinging jet develops two free streamlines, which initiate smoothly at the endpoints of the semi-infinitely long nozzle, and the speed on free streamlines remains a constant, which can be determined by the impinging jet flow itself. The asymptotic behaviors of the oblique impinging jet flows at the far fields, the position of the stagnation points, convexity of the free boundaries and other properties are also considered. On another side, it is showed in this paper that there does not exist an oblique impinging jet flow with one asymptotic direction generally. [ABSTRACT FROM AUTHOR]

In this paper, we consider a domain Ω ε ⊂ R N , N ≥ 2 , with a very rough boundary depending on ε . For instance, if N = 3 Ω ε has the form of a brush with an ε -periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order ε . In Ω ε we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε , on the lateral boundary of the teeth. We study the asymptotic behavior of this problem, as ε vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system. [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]

A huge volume of research has been done for the simplest chemotaxis model (Keller–Segel's minimal model ) and its variants, yet, some of the basic issues remain unresolved until now. For example, it is known that the minimal model has spiky steady states that can be used to model the important cell aggregation phenomenon, but the stability of monotone spiky steady states was not shown. In this paper, we derive, first formally and then rigorously, the asymptotic expansion of these monotone steady states, and then we use this fine information on the spike to prove its local asymptotic stability. Moreover, we obtain the uniqueness of such steady states. We expect that the new ideas and techniques for rigorous asymptotic expansion and spectrum analysis presented in this paper will be useful in attacking and hence stimulating research on other more sophisticated chemotaxis models. [ABSTRACT FROM AUTHOR]

This paper concerns the equation (0.1) u t = u x x + f ( x − c t , u ) , x ∈ R , where c ≥ 0 is a forcing speed and f : ( s , u ) ∈ R × R + → R is asymptotically of KPP type as s → − ∞ . We are interested in the questions of whether such a forced moving KPP nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist. Under a sublinearity condition on f ( s , u ) , we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases. In these cases, we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in s ∈ R . [ABSTRACT FROM AUTHOR]

This paper deals with the following linearly coupled nonlinear Kirchhoff-type system: { − ( a 1 + b 1 ∫ R 3 | ∇ u | 2 d x ) Δ u + μ 1 u = f ( u ) + β v in R 3 , − ( a 2 + b 2 ∫ R 3 | ∇ v | 2 d x ) Δ v + μ 2 v = g ( v ) + β u in R 3 , u , v ∈ H 1 ( R 3 ) , where a i > 0 , b i ≥ 0 , μ i > 0 are constants for i = 1 , 2 , β > 0 is a parameter and f , g ∈ C ( R , R ) . Under the general Berestycki–Lions type assumptions on f and g , we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods. We also study the asymptotic behavior of these solutions as β → 0 + . [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the time periodic Lotka–Volterra competition–diffusion system where are T-periodic functions, , . Under certain conditions, the system admits two stable semi-trivial periodic solutions and and a unique coexistence periodic solution , which is unstable and satisfies and for . In this paper we prove that the system admits a time periodic traveling wave solution connecting two periodic solutions and as , where c is the wave speed. By using a dynamical method, we show that the time periodic traveling wave solution is asymptotically stable and unique modulo translation for front-like initial values. [Copyright &y& Elsevier]

In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result ( Theorem A ) we give a uniform asymptotic expansion of the trajectories arriving at the node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result ( Theorem B ) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a building block in the study of bifurcations of critical periods in a neighborhood of a polycycle. Finally, we apply Theorems A and B to the study of critical periods of the Loud family of quadratic centers and we prove that no bifurcation occurs for certain values of the parameters ( Theorem C ). [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotically linear system − Δ u 1 = λ θ 1 u 3 + f 1 ( λ , x , u 1 , u 2 , u 3 ) in Ω − Δ u 2 = λ θ 2 u 2 + f 2 ( λ , x , u 1 , u 2 , u 3 ) in Ω − Δ u 3 = λ θ 3 u 1 + f 3 ( λ , x , u 1 , u 2 , u 3 ) in Ω u 1 = u 2 = u 3 = 0 on ∂ Ω , } where θ i > 0 for i = 1 , 2 , 3 with θ 2 ≠ θ 1 θ 3 , λ is a real parameter and Ω ⊂ R N is a bounded domain with smooth boundary. The linear part of the system has two simple eigenvalues with nonnegative eigenfunctions each with at least one zero component. We provide sufficient conditions which guarantee bifurcation from infinity of positive solutions from both, one or none of the two simple eigenvalues. Under additional assumptions on the nonlinear perturbations, we determine the λ -direction of bifurcation as well. We use bifurcation theory to establish our results. [ABSTRACT FROM AUTHOR]

The asymptotic expansions of the Navier–Stokes flow in R n and the rates of decay are studied with aid of weighted Hardy spaces. Fujigaki and Miyakawa [12] , Miyakawa [28] proved the n th order asymptotic expansion of the Navier–Stokes flow if initial data decays like ( 1 + | x | ) − n − 1 and if n th moment of initial data is finite. In the present paper, it is clarified that the moment condition for initial data is essential in order to obtain higher order asymptotic expansion of the flow and to consider the rapid time decay problem. The second author [39] established the weighted estimates of the strong solutions in the weighted Hardy spaces with small initial data which belongs to L n and a weighed Hardy space. Firstly, the refinement of the previous work [39] is achieved with alternative proof. Then the existence time of the solution in the weighted Hardy spaces is characterized without any Hardy norm. As a result, in two dimensional case the smallness condition on initial data is completely removed. As an application, the rapid time decay of the flow is investigated with aid of asymptotic expansions and of the symmetry conditions introduced by Brandolese [3] . [ABSTRACT FROM AUTHOR]

Abstract: The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space . In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable . For the variable in the normal direction, we use space or weighted space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for . The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) . [Copyright &y& Elsevier]

Abstract: This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher–KPP reaction–diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the study of the linear damped p-system with boundary effect. By a heuristic analysis, we realize that the best asymptotic profile for the original solution is the parabolic solution of the IBVP for the corresponding porous media equation with a specified initial data. In particular, we further show the convergence rates of the original solution to its best asymptotic profile, which are much better than the existing rates obtained in the previous works. The approach adopted in the paper is the elementary weighted energy method with Green function method together. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole–Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given by the inverse Cole–Hopf transformation of a summation of n heat kernels. The k-th order moments of the exact and the approximate solution are contracting with order in -norm as . This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of semilinear functional evolution equations in which the nonlinearity is demicontinuous and satisfies a semimonotone condition. We prove the existence, uniqueness and exponentially asymptotic stability of the mild solutions. Our approach is to apply a convenient version of Burkholder inequality for convolution integrals and an iteration method based on the existence and measurability results for the functional integral equations in Hilbert spaces. An Itô-type inequality is the main tool to study the uniqueness, p-th moment and almost sure sample path asymptotic stability of the mild solutions. We also give some examples to illustrate the applications of the theorems and meanwhile we compare the results obtained in this paper with some others appeared in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we study a free boundary value problem for two-phase liquid–gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity. This is an extension of the paper [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid–gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV]. Just as in [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid–gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV], the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when , which have improved the previous result of Evje and Karlsen, and get the asymptotic behavior result, also we obtain the regularity of the solutions by energy method. [Copyright &y& Elsevier]

Abstract: In this paper we conclude the analysis started in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551–597] and continued in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174–202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction–diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551–597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174–202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in and norms. [Copyright &y& Elsevier]

Abstract: In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317–318): when a supersonic flow hits a sharp wedge, it follows from the Rankine–Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined. [Copyright &y& Elsevier]

Abstract: In this paper, we study small polynomial perturbations of a Hamiltonian vector field with Hamiltonian F formed by a product of real linear functions in two variables. We assume that the corresponding lines are in a general position in . That is, the lines are distinct, non-parallel, no three of them have a common point and all critical values not corresponding to intersections of lines are distinct. We prove in this paper that the principal Poincaré–Pontryagin function , associated to such a perturbation and to any family of ovals surrounding a singular point of center type, belongs to the -module generated by Abelian integrals and some integrals , with defined in the paper. Moreover, are not Abelian integrals. They are iterated integrals of length two. [Copyright &y& Elsevier]

Abstract: In this paper we study the period function of centers of planar polynomial differential systems. With a convenient compactification of the phase portrait, the boundary of the period annulus of the center has two connected components: the center itself and a polycycle. We are interested in the behaviour of the period function near the polycycle. The desingularization of its critical points gives rise to a new polycycle (monodromic as well) with hyperbolic saddles or saddle-nodes at the vertices. In this paper we compute the first terms in the asymptotic development of the time function around any orbitally linearizable saddle that may come from this desingularization process. In addition, we use these developments to study the bifurcation diagram of the period function of the dehomogenized Loud''s centers. More generally, the tools developed here can be used to study the return time function around a monodromic polycycle. This work is a continuation of the results in [P. Mardešić, D. Marín, J. Villadelprat, On the time function of the Dulac map for families of meromorphic vector fields, Nonlinearity 16 (2003) 855–881; P. Mardešić, D. Marín, J. Villadelprat, The period function of reversible quadratic centers, J. Differential Equations 224 (2006) 120–171]. [Copyright &y& Elsevier]

This paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary. It is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar, the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic. This paper is devoted to generalizing this result. Under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small, a sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to a global weak solution of the steady problem. A sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution. The regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied. [Copyright &y& Elsevier]

This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem. [ABSTRACT FROM AUTHOR]

This paper examines a class of linear hyperbolic systems which generalizes the Goldstein–Kac model to an arbitrary finite number of speeds v i with transition rates μ i j . Under the basic assumptions that the transition matrix is symmetric and irreducible, and the differences v i − v j generate all the space, the system exhibits a large-time behavior described by a parabolic advection–diffusion equation. The main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters v i and μ i j , establishing a complete connection between microscopic and macroscopic coefficients. It is shown that the drift speed is the arithmetic mean of the velocities v i . The diffusion matrix has a more complicate representation, based on the graph with vertices the velocities v i and arcs weighted by the transition rates μ i j . The approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchoff's matrix tree Theorem from graph theory. [ABSTRACT FROM AUTHOR]

We investigate an optimal control problem with an averaging cost. The asymptotic behaviour of the values is a classical problem in ergodic control. To study the long run averaging we consider both Cesàro and Abel means. A main result of the paper says that there is at most one possible accumulation point – in the uniform convergence topology – of the values, when the time horizon of the Cesàro means converges to infinity or the discount factor of the Abel means converges to zero. This unique accumulation point is explicitly described by representation formulas involving probability measures on the state and control spaces. As a byproduct we obtain the existence of a limit value whenever the Cesàro or Abel values are equicontinuous. Our approach allows to generalise several results in ergodic control, and in particular it allows to cope with cases where the limit value is not constant with respect to the initial condition. [ABSTRACT FROM AUTHOR]

Consider a continuous-time linear switched system on R n associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a Barabanov norm associated with the system. This paper deals with two types of issues: ( a ) properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; ( b ) asymptotic behavior of the extremal solutions of the linear switched system. Regarding Issue ( a ), we provide partial answers and propose four related open problems. As for Issue ( b ), we establish, when n = 3 , a Poincaré–Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behavior of linear switched system on R 3 associated with a pair of Hurwitz matrices { A , A + b c T } . After pointing out a gap in Barabanov's proof we partially recover his result by alternative arguments. [ABSTRACT FROM AUTHOR]

This short paper concerns a diffusive logistic equation with a free boundary and sign-changing coefficient, which is formulated to study the spread of an invasive species, where the free boundary represents the expanding front. A spreading–vanishing dichotomy is derived, namely the species either successfully spreads to the right-half-space as time t → ∞ and survives (persists) in the new environment, or it fails to establish itself and will extinct in the long run. The sharp criteria for spreading and vanishing are also obtained. When spreading happens, we estimate the asymptotic spreading speed of the free boundary. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the asymptotic behavior of boundary blow-up solutions to the equation in Ω, where is the ∞-Laplacian, the nonlinearity f is a positive, increasing function in , and the weighted function is positive in Ω and may vanish on the boundary. We first establish the exact boundary blow-up estimates with the first expansion when f is regularly varying at infinity with index and the weighted function b is controlled on the boundary in some manner. Furthermore, for the case of , with the function g normalized regularly varying with index , we obtain the second expansion of solutions near the boundary. It is interesting that the second term in the asymptotic expansion of boundary blow-up solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the boundary blow-up problems involving the classical Laplacian. [Copyright &y& Elsevier]

Abstract: The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, . The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders. [Copyright &y& Elsevier]