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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]

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]

Abstract: This paper is concerned with asymptotic profiles of ground states for a class of quasilinear Schrödinger equations with -supercritical nonlinearities. We study the blow-up rate of ground states by using the dual variational structure of equations as well as various variational methods. [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: In this paper we consider a free boundary problem for a reaction–diffusion logistic equation with a time-dependent growth rate. Such a problem arises in the modeling of information diffusion in online social networks, with the free boundary representing the spreading front of news among users. We present several sharp thresholds for information diffusion that either lasts forever or suspends in finite time. In the former case, we give the asymptotic spreading speed which is determined by a corresponding elliptic equation. [Copyright &y& Elsevier]

Abstract: In this paper we exhibit the dissipative mechanism of the Cahn–Hilliard equation in . We show a weak form of dissipativity by showing that each individual solution is attracted, in some sense, by the set of equilibria. We also indicate that strong dissipativity, that is, asymptotic compactness in , cannot be in general expected. Then we consider two types of perturbations: a nonlinear perturbation and a small linear perturbation. In both cases we show that, for the resulting equations, the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor. [Copyright &y& Elsevier]

Abstract: In this paper we study radially symmetric entire solutions of We find the asymptotic behavior of these solutions for any and prove that for any there exists a solution with exactly linear growth. [Copyright &y& Elsevier]

Abstract: We study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) , and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) . In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) and Du and Guo (2011) to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here. [Copyright &y& Elsevier]

Abstract: We treat a class of equations given by , , , and obtain rigorous uniform asymptotic expansions of the solutions as . A key tool is a new formula of variation of constants that works for such quadratic equations. Included are solutions with one or more spikes. One example of this class of problems is a famous problem studied by Carrier and discussed formally by Bender and Orszag in the book “Advanced Mathematical Methods for Scientists and Engineers”. In this paper we give the first rigorous derivation for the well-known asymptotics for that problem. Another more applied example is also covered by our theory. [Copyright &y& Elsevier]

Abstract: In this paper we consider zero-relaxation limits for periodic smooth solutions of Euler–Maxwell systems. For well-prepared initial data, we propose an approximate solution based on a new asymptotic expansion up to any order. For ill-prepared initial data, we construct initial layer corrections in an explicit way. In both cases, the asymptotic expansions are valid in time intervals independent of the relaxation time and their convergence is justified by establishing uniform energy estimates. [Copyright &y& Elsevier]

Abstract: We study large-time asymptotics for a class of noncoercive Hamilton–Jacobi equations with Dirichlet boundary condition in one space dimension. We prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset. We show that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. This work is an extension with more rigorous analysis of a recent paper by E. Yokoyama, Y. Giga and P. Rybka, in which a growing crystal model is established and the asymptotic behavior described above is first discovered. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a semilinear elliptic boundary value problem in a smooth bounded domain, having the so-called logistic nonlinearity that originates from population dynamics, with a nonlinear boundary condition. Although the logistic nonlinearity has an absorption effect in the problem, the nonlinear boundary condition is induced by the homogeneous incoming flux on the boundary. The objective of our study is to analyze the existence of a bifurcation component of positive solutions from trivial solutions and its asymptotic behavior and stability. We perform this analysis using the method developed by Lyapunov and Schmidt, based on a scaling argument. [Copyright &y& Elsevier]

Abstract: In this paper we address an initial-value problem for the defocusing modified Korteweg–de Vries (mKdV−) equation. The normalized modified Korteweg–de Vries equation considered is given by where x and τ represent dimensionless distance and time respectively and is a constant. We consider the case when the initial data has a discontinuous step, where for and for . The method of matched asymptotic coordinate expansions is used to obtain the complete large-τ asymptotic structure of the solution to this problem, which exhibits the formation of a permanent form travelling wave (kink) solution propagating in the −x direction with speed and connecting to , while the solution is oscillatory in as (oscillating about ), with the oscillatory envelope being of as . The asymptotic correction to the propagation speed of the travelling wave solution is given by as , and the rate of convergence of the solution of the initial-value problem to the travelling wave solution is found to be algebraic in τ, as , being of . A brief discussion of the structure of the large-time solution to the mKdV− equation when the initial data is given by the general discontinuous step, for and for , is also given. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to justify the Stokes–Blasius law of boundary-layer thickness for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane, i.e., we shall prove that the boundary-layer thickness is of the value with any for small diffusivity coefficient . Moreover, the convergence rates of the vanishing diffusivity limit are also obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of positive solutions and sign-changing solutions for the following boundary value problem in the half-space where λ is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically linear at infinity. [ABSTRACT FROM AUTHOR]

Abstract: In the early 2000''s, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov–Petrovskii–Piskunov–Fisher equation Since then, this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction–diffusion equations. In this paper, we give a complete solution to the problem of existence and uniqueness of monotone waves in Eq. (⁎). We show that each monotone traveling wave can be found via an iteration procedure. The proposed approach is based on the use of special monotone integral operators (which are different from the usual Wu–Zou operator) and appropriate upper and lower solutions associated to them. The analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller–Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates. [Copyright &y& Elsevier]