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In this paper, we study the cost of null controllability of a fourth-order parabolic equation. When the control time is large enough, we prove that this cost decreases exponentially to zero as the diffusion coefficient of the equation vanishes. When the control time is small, on the contrary, we prove that this cost increases exponentially to infinity. [ABSTRACT FROM AUTHOR]

In this paper we consider an energy critical wave equation ( 3 ≤ d ≤ 5 , ζ = ± 1 ) ∂ t 2 u − Δ u = ζ ϕ ( x ) | u | 4 / ( d − 2 ) u , ( x , t ) ∈ R d × R with initial data ( u , ∂ t u ) | t = 0 = ( u 0 , u 1 ) ∈ H ˙ 1 × L 2 ( R d ) . Here ϕ ∈ C ( R d ; ( 0 , 1 ] ) converges as | x | → ∞ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ∂ t 2 u − Δ u = | u | 4 / ( d − 2 ) u . Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case ζ = − 1 . While in the focusing case ζ = 1 we give a criterion for global behaviour of the solutions, either scattering or finite-time blow-up when the energy is smaller than a certain threshold. As an application we give a similar criterion on the global behaviour of radial solutions to the focusing, energy critical shifted wave equation ∂ t 2 v − ( Δ H 3 + 1 ) v = | v | 4 v on the hyperbolic space H 3 . [ABSTRACT FROM AUTHOR]

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of d -dimensional critical fractal Burgers equation u t − Δ α / 2 u + b ⋅ ∇ ( u | u | q ) = 0 , α ∈ ( 1 , 2 ) , b ∈ R d for q = ( α − 1 ) / d and u 0 ∈ L 1 ( R d ) . We consider also q > ( α − 1 ) / d under additional condition u 0 ∈ L ∞ ( R d ) . In both cases we assume u 0 ≥ 0 , which implies that the solution is non-negative. The estimates are given in the terms of the function P t u 0 , where P t denotes the semigroup for the operator ∂ t − Δ α / 2 . [ABSTRACT FROM AUTHOR]

In this paper, we prove the solution map of the Cauchy problem of Camassa–Holm type equations depends continuously on the initial data in nonhomogeneous Besov spaces in the sense of Hadamard by using the Littlewood–Paley theory and the method introduced by Kato [37] and Danchin [21] . [ABSTRACT FROM AUTHOR]

In this paper we study the dynamics of parabolic semilinear differential equations with homogeneous Dirichlet boundary conditions via the discretization of finite element method. We provide an appropriate functional setting to treat this problem and, as a first step, we show the continuity of the set of equilibria and of its linear unstable manifolds. [ABSTRACT FROM AUTHOR]

Classical theories predict that solutions of differential equations will leave any neighborhood of a stable limit cycle, if white noise is added to the system. In reality, many engineering systems modeled by second order differential equations, like the van der Pol oscillator, show incredible robustness against noise perturbations, and the perturbed trajectories remain in the neighborhood of a stable limit cycle for all times of practical interest. In this paper, we propose a new model of noise to bridge this apparent discrepancy between theory and practice. Restricting to perturbations from within this new class of noise, we consider stochastic perturbations of second order differential systems that –in the unperturbed case– admit asymptotically stable limit cycles. We show that the perturbed solutions are globally bounded and remain in a tubular neighborhood of the underlying deterministic periodic orbit. We also define stochastic Poincaré map(s), and further derive partial differential equations for the transition density function. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of solutions for the following elliptic problem, related to mean-field games systems: { − div ( M ( x ) ∇ ζ ) + ζ − div ( ζ A ( x ) ∇ u ) = f in Ω , − div ( M ( x ) ∇ u ) + u + θ A ( x ) ∇ u ⋅ ∇ u = ζ p in Ω , ζ = 0 = u on ∂ Ω , where p > 0 , 0 < θ < 1 , and f ≥ 0 is a function in some Lebesgue space. [ABSTRACT FROM AUTHOR]

In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Łojasiewicz–Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated L 2 -gradient flow converges for large time to an elastica, that is to a critical point of the functional. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new estimates for the trace and inverse trace of Steklov eigenvalues. The estimates generalize some previous results of Hersch–Payne–Schiffer [13] , Brock [2] , Raulot–Savo [21] and Dittmar [5] . [ABSTRACT FROM AUTHOR]

Abstract: A continuous ordinary vector differential equation in Euclidean space has a funnel of solutions through each initial condition. Its cross-section at time t is a continuum. Many continua are known to be funnel sections: For instance the circle is a cross-section of a continuous ODE where y is a variable in the plane, but it is not known whether every Jordan curve J is a planar funnel section. In this paper we give sufficient conditions that imply J is a planar funnel section – “pierceability.” We show that pierceability is not generic when we put a fairly interesting complete metric on the space of Jordan curves. We also give proofs of several statements in the first authorʼs paper on funnel sections that appeared in the JDE in 1975. [Copyright &y& Elsevier]

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

Abstract: This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the -well-posedness which is also necessary if the space dimension is equal to one. The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix. [Copyright &y& Elsevier]

Abstract: This paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random “mixture” of self-similar Koch curves. We construct an ε-thin polygonal 2-dimensional fiber , , , around pre-fractal approximating domains and related suitable energy functionals. The aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while, simultaneously, the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as . [Copyright &y& Elsevier]

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

Abstract: This paper is concerned with the nonlinear full Marguerre–von Kármán shallow shell system with a dissipative mechanism of memory type. The model depends on one small parameter. The main purpose of this paper is to show that as the parameter approaches zero, the limiting system is the well-known full von Kármán model with memory for thin plates. [ABSTRACT FROM AUTHOR]

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

Abstract: In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier–Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier–Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483–500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method. [Copyright &y& Elsevier]

Abstract: In a recent paper by D. Peralta-Salas in this journal (J. Differential Equations 244 (6) (2008) 1287–1303) there appears an incorrect result relating symmetries and first integrals of a vector field. The proof relies on a nonexistent theorem in a paper by Sherring and Prince (Trans. Amer. Math. Soc. 334 (1992) 433–453); the error is corrected in this comment. [Copyright &y& Elsevier]

Abstract: Let X be a Banach space whose dual space is uniformly convex. We demonstrate that, for any demicontinuous, weakly Nagumo, k-pseudo-contractive mapping with closed domain, weakly generates a semigroup on . In this paper, we project the consequences of this result on fixed point theory. In particular, we show that if (id est, if T is strongly pseudo-contractive), then T has a unique fixed point. This implies that, if T is pseudo-contractive () and is closed, bounded, and convex, then T has at least one fixed point. Consequently, any demicontinuous pseudo-contractive mapping (for an appropriate C) has a fixed point, which has been an important open question in fixed point theory for quite some time. In a subsequent paper, we explore the consequences of the semigroup result on the existence of solutions to certain partial differential equations. The semigroup result directly implies the existence of unique global solutions to time evolution equations of the form where A is a combination of derivatives. The fixed point results from this paper imply the existence of solutions to partial differential equations of the form . [Copyright &y& Elsevier]

Abstract: In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243–260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515–1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094–1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu–Yang functional, one can prove the stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226–1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098] to a more general case. [Copyright &y& Elsevier]

Abstract: The dynamics of dilute electrons and plasma can be modeled by Vlasov–Poisson–Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is , by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245–316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro–micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components. [Copyright &y& Elsevier]

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

Abstract: We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis–Procesi equation In a recent paper [G.M. Coclite, K.H. Karlsen, On the well-posedness of the Degasperis–Procesi equation, J. Funct. Anal. 233 (2006) 60–91], we proved for this equation the existence and uniqueness of weak solutions satisfying an infinite family of Kružkov-type entropy inequalities. The purpose of this paper is to replace the Kružkov-type entropy inequalities by an Oleĭnik-type estimate and to prove uniqueness via a nonlocal adjoint problem. An implication is that a shock wave in an entropy weak solution to the Degasperis–Procesi equation is admissible only if it jumps down in value (like the inviscid Burgers equation). [Copyright &y& Elsevier]

Abstract: In this paper we prove that the spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof. [Copyright &y& Elsevier]

Abstract: We consider a 2mth order elliptic operator of divergence form in a domain Ω of , whose leading coefficients are uniformly continuous. In the paper [Y. Miyazaki, The theory of divergence form elliptic operators under the Dirichlet condition, J. Differential Equations 215 (2005) 320–356], we developed the theory including the construction of resolvents, assuming that the boundary of Ω is of class . The purpose of this paper is to show that the theory also holds when Ω is a domain, applying the inequalities of Hardy type for the Sobolev spaces. [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]

Abstract: The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation. [Copyright &y& Elsevier]

Abstract: In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationwhere the nonlinear term depends on the unknown function u and its derivative and satisfy the estimatewith such that The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions. [Copyright &y& Elsevier]

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

Abstract: In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states at a nonlinear eigenvalue with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz''s unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz''s context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181). [Copyright &y& Elsevier]

This paper is concerned with the asymptotic behavior of solution to the Cahn–Hilliard equation subject to the following dynamic boundary conditions: and the initial condition where Ω is a bounded domain in Rn (n⩽3) with smooth boundary Γ , and Γs>0, σs>0, gs>0, hs are given constants; Δ|| is the tangential Laplacian operator, and ν is the outward normal direction to the boundary.This problem has been considered in the recent paper by Racke and Zheng (Adv. Differential Equations 8 (1) (2003) 83) where the global existence and uniqueness were proved. In a very recent manuscript by Prüss, Racke and Zheng (Konstanzer Schrift. Math. Inform. 189 (2003)) the results on existence of global attractor and maximal regularity of solution have been obtained. In this paper, convergence of solution of this problem to an equilibrium, as time goes to infinity, is proved. [Copyright &y& Elsevier]

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics. [Copyright &y& Elsevier]

This paper is a continuation of the previous paper (J. Differential Equations 165 (2000) 255). The main subject is the Gevrey property of formal solutions of an analytic ordinary differential equation in powers of a parameter. In one case, a given formal solution itself is of the Gevrey type, while, in another case, the existence of a formal solution implies the existence of formal solutions of the Gevrey types. These situations are explained systematically in this paper. [Copyright &y& Elsevier]

In this paper we derive a general limit characterization of principal solutions at infinity of linear Hamiltonian systems under no controllability assumption. The main result is formulated in terms of a limit involving antiprincipal solutions at infinity of the system. The novelty lies in the fact that the principal and antiprincipal solutions at infinity may belong to two different genera of conjoined bases, i.e., the eventual image of their first components is not required to be the same as in the known literature. For this purpose we extend the theory of genera of conjoined bases, which was recently initiated by the authors. We show that the orthogonal projector representing each genus of conjoined bases satisfies a symmetric Riccati matrix differential equation. This result then leads to an exact description of the structure of the set of all genera, in particular it forms a complete lattice. We also provide several examples, which illustrate our new theory. [ABSTRACT FROM AUTHOR]

In this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point. We prove the existence of many complicated dynamical objects for a large class of non-autonomous second order equations, ranging from attractive quasi-periodic torus to Newhouse sinks and Hénon-like attractors, and to rank one attractors with SRB measures and full stochastic behavior. This theory enables us to apply rigorously many profound dynamics theories on non-uniformly hyperbolic maps developed in the last forty years, including the Newhouse theory, the theory of SRB measures, the theory of Hénon-like attractors and the theory of rank one attractors, to the analysis of the strange attractors in a periodically perturbed Duffing equation. [ABSTRACT FROM AUTHOR]

Consider control systems described by a differential equation with a control term or, more generally, by a differential inclusion with velocity set F ( t , x ) . Certain properties of state trajectories can be derived when it is assumed that F ( t , x ) is merely measurable w.r.t. the time variable t . But sometimes a refined analysis requires the imposition of stronger hypotheses regarding the time dependence. Stronger forms of necessary conditions for minimizing state trajectories can be derived, for example, when F ( t , x ) is Lipschitz continuous w.r.t. time. It has recently become apparent that significant addition properties of state trajectories can still be derived, when the Lipschitz continuity hypothesis is replaced by the weaker requirement that F ( t , x ) has bounded variation w.r.t. time. This paper introduces a new concept of multifunctions F ( t , x ) that have bounded variation w.r.t. time near a given state trajectory, of special relevance to control. We provide an application to sensitivity analysis. [ABSTRACT FROM AUTHOR]

This paper gives a classification of polynomial differential operators X = X 1 ( x 1 , x 2 ) δ 1 + X 2 ( x 1 , x 2 ) δ 2 ( δ i = ∂ / ∂ x i ) . The classification is defined through an order derived from X . Let X = X y be the associated differential polynomial, the order is defined as the order of a differential ideal Λ that is an essential extension of { X } . The main result shows the order can only be four possible values: 0, 1, 2, 3, or ∞. Furthermore, when the order is finite, the essential extension Λ = { X , A } , where A is a differential polynomial with coefficients obtained through a rational solution of a partial differential equation given explicitly by coefficients of X . When the order is infinite, the extension Λ is identical with { X } . In addition, if, and only if, the order is 0, 1, or 2, the associated polynomial differential equation has Liouvillian first integrals. Examples and connections with Godbillon–Vey sequences are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we study the following Hénon-like equation { − Δ u = | | y | − 2 | α u p , u > 0 , in Ω , u = 0 , on ∂ Ω , where α > 0 , p = N + 2 N − 2 , Ω = { y ∈ R N : 1 < | y | < 3 } , N ≥ 4 . We show that for α > 0 the above problem has infinitely many positive solutions concentrating simultaneously near the interior boundary { x ∈ R N : | x | = 1 } and the outward boundary { x ∈ R N : | x | = 3 } , whose energy can be made arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of traveling waves to a class of lattice differential equations for cellular neural networks with multiple delays. Following the previous study [38] on the existence of the traveling waves, here we focus on the uniqueness and the stability of these traveling waves. First of all, by establishing the a priori asymptotic behavior of traveling waves and applying Ikehara's theorem, we prove the uniqueness (up to translation) of traveling waves ϕ ( n − c t ) with c ≤ c ⁎ for the cellular neural networks with multiple delays, where c ⁎ < 0 is the critical wave speed. Then, by the weighted energy method together with the squeezing technique, we further show the global stability of all non-critical traveling waves for this model, that is, for all monotone waves with the speed c < c ⁎ , the original lattice solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations around the monotone traveling waves decay exponentially at far fields, but can be arbitrarily large in other locations. [ABSTRACT FROM AUTHOR]

In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7] . [ABSTRACT FROM AUTHOR]

In this paper we analyze operator splitting for the Benjamin–Ono equation, u t = u u x + H u x x , where H denotes the Hilbert transform. If the initial data are sufficiently regular, we show the convergence of both Godunov and Strang splittings. [ABSTRACT FROM AUTHOR]

In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than Hölder, namely bounded coefficients. As for second order equations in [14] we prove that such equations admit a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter. Classical solutions are recovered as limit of very weak solutions. Finally, by using a reduction to block Sylvester form we conclude that any first order hyperbolic system with non-regular coefficients is solvable in the very weak sense. [ABSTRACT FROM AUTHOR]

This paper is devoted to an integrable two-component Camassa–Holm system with cubic nonlinearity, which includes the cubic Camassa–Holm equation (also called the Fokas–Olver–Rosenau–Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data. [ABSTRACT FROM AUTHOR]

Abstract evolution differential equations of second order in time are studied in order to get (almost) optimal decay estimates to the corresponding energy functional of the equations. The framework is supported by a special energy method in the associated Fourier space. The constructed abstract theory can be applied to several concrete evolutionary partial differential equations as is illustrated in the last section of the paper. [ABSTRACT FROM AUTHOR]

This paper studies the equivalence between differentiable and non-differentiable dynamics in R n . Filippov's theory of discontinuous differential equations allows us to find flow solutions of dynamical systems whose vector fields undergo switches at thresholds in phase space. The canonical convex combination at the discontinuity is only the linear part of a nonlinear combination that more fully explores Filippov's most general problem: the differential inclusion. Here we show how recent work relating discontinuous systems to singular limits of continuous (or regularized ) systems extends to nonlinear combinations. We show that if sliding occurs in a discontinuous systems, there exists a differentiable slow–fast system with equivalent slow invariant dynamics. We also show the corresponding result for the pinching method, a converse to regularization which approximates a smooth system by a discontinuous one. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of positive radial solutions of the scalar curvature equation, i.e. Δ u ( x ) + K ( | x | ) u σ − 1 ( x ) = 0 where σ = 2 n n − 2 and we assume that K ( | x | ) = k ( | x | ε ) and k ( r ) ∈ C 1 is bounded and ε > 0 is small. It is known that we have at least a ground state with fast decay for each positive critical point of k for ε small enough. In fact if the critical point k ( r 0 ) is unique and it is a maximum we also have uniqueness; surprisingly we show that if k ( r 0 ) is a minimum we have an arbitrarily large number of ground states with fast decay. The results are obtained using Fowler transformation and developing a dynamical approach inspired by Melnikov theory. We emphasize that the presence of subharmonic solutions arising from zeroes of Melnikov functions has not appeared previously, as far as we are aware. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the local rapid exponential stabilization problem for a controlled Kuramoto–Sivashinsky equation on a bounded interval. We build a feedback control law to force the solution of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates, provided that the initial datum is small enough. Our approach uses a method we introduced for the rapid stabilization of a Korteweg–de Vries equation. It relies on the construction of a suitable integral transform and can be applied to many other equations. [ABSTRACT FROM AUTHOR]

We consider the following Chern–Simons equation, (0.1) Δ u + 1 ε 2 e u ( 1 − e u ) = 4 π ∑ i = 1 N δ p i ε , in Ω , where Ω is a 2-dimensional flat torus, ε > 0 is a coupling parameter and δ p stands for the Dirac measure concentrated at p . In this paper, we proved that the topological solutions of (0.1) are uniquely determined by the location of their vortices provided the coupling parameter ε is small and the collapsing velocity of vortices p i ε is slow enough or fast enough comparing with ε . This extends the uniqueness results of Choe [5] and Tarantello [22] . Meanwhile, for any topological solution ψ defined in R 2 whose linearized operator is non-degenerate, we construct a sequence of topological solutions u ε of (0.1) whose asymptotic limit is exactly ψ after rescaling around 0. A consequence is that non-uniqueness of topological solutions in R 2 implies non-uniqueness of topological solutions on torus with collapsing vortices. [ABSTRACT FROM AUTHOR]

We discussed in a previous paper elliptic systems of partial differential equations on star-shaped domains and introduced the notions of conjugate radius and bifurcation radius. We proved that every bifurcation radius is a conjugate radius, and believed to have shown by an example that on the other hand not every conjugate radius is a bifurcation radius. This note reveals that our previous example was wrong, but it also introduces an improved example that shows the assertion that we claimed before. [ABSTRACT FROM AUTHOR]