pinecone/core-2020-05-10-deduplication huggingface dataset:
The Successive Derivatives of the Period Function of a Plane Vector Field

AbstractPreviously, we provided an expression which generalized the classical Melnikov function to any order, for the first nonzero derivative of a return mapping. Our method relied on the decomposition of a 1-form associated to the relative cohomology of the perturbed Hamiltonian. With the same techniques, we give a formula for the first nonzero derivative of a period function. We focus on the particular example ofH=(1/2)(x2+y2) and then we define a class of Hamiltonians for which the same computation remains valid. Finally, we investigate relations with Birkhoff normal form
The Behavior of the Life Span for Solutions tout=Δu+a(x)upinRd

AbstractLetT*(λ,φ) denote the life span of the positive, bounded solutionu(x,t) to the equationut=Δu+a(u)upinRdandu(x,0)=λφ(x), where 0≨a(x)∈Cα(Rd), 0≨φ(x)∈Cb(Rd),p>1, andλ>0 is a parameter. Depending ona,φ,p, andd, it is possible thatT*(λ,φ)=∞, forλ>0 sufficiently small, or thatT*(λ,φ)<∞, for allλ>0, in which case limλ→0T*(λ,φ)=∞. It is always true that limλ→∞T*(λ,φ)=0. In this paper we investigate the asymptotic behavior ofT*(λ,φ) asλ→0 in the case thatT*(λ,φ)<∞, for allλ>0, and asλ→∞ in all cases. The asymptotic order depends heavily ona,φ,p, anddin the case thatλ→0, whereas in the case thatλ→∞, it depends only on whether there exists anx0witha(x0),φ(x0)≠0, or whether the supports ofaandφare separated by a positive distance
Existence and Relaxation Results for Nonlinear Second-Order Multivalued Boundary Value Problems in RN

AbstractIn this paper we study second order differential inclusions with nonlinear boundary conditions. Our formulation is general and incorporates as special cases well-known problems such as the Dirichlet (Picard), Neumann, and periodic problems. We prove existence theorems under various sets of hypotheses for both the convex and nonconvex problems. Also we show the existence of extremal solutions and that the extremal solutions are dense in the solutions of the convex problem for theW1,2(T,RN)-norm (strong relaxation theorem). Finally we examine the Dirichlet problem when the multivalued right-hand side does not depend on the derivative of x and satisfies a general growth hypothesis and a sign-type condition. For this problem we prove existence results and a relaxation theorem
A Priori Bounds and Multiple Solutions for Superlinear Indefinite Elliptic Problems

AbstractIn this work we study existence and multiplicity questions for positive solutions of second-order semilinear elliptic boundary value problems, where the nonlinearity is multiplied by a weight function which is allowed to change sign and vanish on sets of positive measure. We do not impose a variational structure, thus techniques from the calculus of variations are not applicable. Under various qualitative assumptions on the nonlinearity we establish a priori bounds and employ bifurcation and fixed point index theory to prove existence and multiplicity results for positive solutions. In an appendix we derive interiorLp-estimates for general elliptic systems of arbitrary order under minimal smoothness hypotheses. Special instances of these results are used in the derivation of a priori bounds
Nonlinear Resonance in Asymmetric Oscillators

AbstractWe study the periodic solutions of equations with asymmetric nonlinearities “at resonance” with the Fučı&#x0301;k spectrum. We compute the associated topological degree and prove existence, multiplicity, and stability of large-amplitude oscillations for equations with a small friction term. Such equations can be viewed, e.g., as simple models for investigating vertical oscillations of long-span suspension bridges. The results are typically of a nonlinear nature, as some of the situations observed cannot occur with a linear equation
On the Relation between Discounted and Average Optimal Value Functions

AbstractWe investigate the relation between discounted and average deterministic optimal control problems for nonlinear control systems. In particular we are interested in the corresponding optimal value functions. Using the concepts of Viability, Chain Controllability, and Controllability, a global convergence result for vanishing discount rate is obtained. Basic ingredients for the analysis are an Abelian type theorem, controllability properties of the system, and the Morse decomposition of the corresponding control flow
On the Integrability of Homogeneous Scalar Evolution Equations

AbstractWe determine the existence of (infinitely many) symmetries for equations of the formut=uk+f(u, …, uk−1)when they areλ-homogeneous (with respect to the scalinguk↦λ+k) withλ>0. Algorithms are given to determine whether a system has a symmetry (also independent oftandx). If it has one generalized symmetry, we prove it has infinitely many and these can be found using recursion operators or master symmetries. The method of proof uses the symbolic method and results from diophantine approximation theory. We list the 10 integrable hierarchies. The methods can in principle be applied to theλ⩽0 case, as we illustrate for one example withλ=0, which seems to be new. In principle they can also be used for systems of evolution equations, but so far this has only been demonstrated for one class of examples
Fractal Burgers Equations

AbstractThe paper studies local and global in time solutions to a class of multidimensional generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and with general algebraic nonlinearity. Such equations naturally appear in continuum mechanics. Our results include existence, uniqueness, regularity and asymptotic behavior of solutions to the Cauchy problem as well as a construction of self-similar solutions. The role of critical exponents is also explained
Scaling Variables and Asymptotic Expansions in Damped Wave Equations

AbstractWe study the long time behavior of small solutions to the nonlinear damped wave equationεuττ+uτ=(a(ξ)uξ)ξ+N(u,uξ,uτ),ξ∈R,τ⩾0, whereεis a positive, not necessarily small parameter. We assume that the diffusion coefficienta(ξ) converges to positive limitsa±asξ→±∞, and that the nonlinearity N(u,uξ,uτ) vanishes sufficiently fast asu→0. Introducing scaling variables and using various energy estimates, we compute an asymptotic expansion of the solutionu(ξ,τ) in powers ofτ−1/2asτ→+∞, and we show that this expansion is entirely determined, up to the second order, by a linear parabolic equation which depends only on the limiting valuesa±. In particular, this implies that the small solutions of the damped wave equation behave for largeτlike those of the parabolic equation obtained by settingε=0
Ergodic Properties and Rotation Number for Linear Hamiltonian Systems

AbstractThis paper is concerned with the dynamical behavior of the solutions of a class of linear Hamiltonian systems, including those to which Kotani's theory applies. We first present a symplecticL2Perron transformation which takes these systems into skew-symmetric form. This allows us to study the average of the trajectories and the Fourier coefficients of the solutions. In addition, from the construction of two invariant complex Lagrange planes, the differentiability of the rotation number is analyzed
On Growth Rates of Subadditive Functions for Semiflows

AbstractLetφ:X×T+→Xbe a semiflow on a compact metric spaceX. A functionF:X×T+→Xis subadditive with respect toφifF(x, t+s)⩽F(x, t)+F(φ(x, t),nbsp;s). We define the maximal growth rate ofFto be supx∈Xlimsupt→∞(1/t)F(x, t). This growth rate is shown to equal the maximal growth rate of the subadditive function restricted to the minimal center of attraction of the semiflow. Applications to Birkhoff sums, characteristic exponents of linear skew-product semiflows on Banach bundles, and average Lyapunov functions are developed. In particular, a relationship between the dynamical spectrum and the measurable spectrum of a linear skew-product flow established by R. A. Johnson, K. J. Palmer, and G. R. Sell (SIAM J. Math. Anal.18, 1987, 1–33) is extended to semiflows in an infinite dimensional setting
On Moments of Negative Eigenvalues for the Pauli Operator

AbstractThis paper concerns the three-dimensional Pauli operatorP=(σ·(p−A(x)))2+V(x)with a nonhomogeneous magnetic fieldB=curlA. The following Lieb–Thirring type inequality for the moment of negative eigenvalues is established as∑λj<0|λj|⩽C1∫R3|V(x)|5/2−dx+C2∫R3[bp(x)]3/2|V(x)|−dx,wherep>3/2 andbp(x) is theLpaverage of |B| over certain cube centered atxwith a side length scaling like |B|−1/2. We also show that, ifBhas a constant direction,∑λj<0|λj|⩽C1,λ∫R3|V(x)|γ+3/2−dx+C2,γ∫R3bp(x)|V(x)|γ+1/2−dx,whereγ>1/2 andp>1
The Floquet Theory of the Periodic Euler–Bernoulli Equation

AbstractWe continue the study (initiated in [17]) of the spectral theory of the fourth-order eigenvalue problem[a(x)u″(x)]″=λρ(x)u(x),−∞<x<∞,where the functionsaandρare periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described byaandρ. The equality of the algebraic and geometric multiplicities of the periodic and antiperiodic eigenvalues is established. Also a spectrum-like set, that we called “pseudospectrum” or “ψ-spectrum,” is introduced (or, rather, discovered). Thisψ-spectrum is shown to lie on the negative real axis and have a band-gap structure. Various open questions and conjectures are mentioned at the end of the paper
Asymptotic Stability and Smooth Lyapunov Functions

AbstractWe establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth Lyapunov function. Since well-known concepts of generalized solutions of differential equations with discontinuous right-hand side can be described in terms of solutions of certain related differential inclusions involving upper semicontinuous multifunctions, this result gives a Lyapunov characterization of asymptotic stability of either Filippov or Krasovskii solutions for differential equations with discontinuous right-hand side. In the study ofweak(as opposed to strong) asymptotic stability, the existence of a smooth Lyapunov function is rather exceptional. However, the methods employed in treating the strong case of asymptotic stability are applied to yield a necessary condition for the existence of a smooth Lyapunov function for weakly asymptotically stable differential inclusions; this is an extension to the context of Lyapunov functons of Brockett's celebrated “covering condition” from continuous feedback stabilization theory
Traveling Waves in Lattice Dynamical Systems

AbstractIn this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODEs) and in coupled map lattices (CMLs). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as time-discretizations of lattice ODEs. Finally, we show that these results apply to the discrete Nagumo equation
Boundary Regularity of Weak Solutions of the Navier–Stokes Equations

AbstractWe prove that a solution to Navier–Stokes equations is inL2(0,∞:H2(Ω)) under the critical assumption thatu∈Lr,r′, 3/r+2/r′⩽1 withr⩾3. A boundaryL∞estimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate forL∞-norm ofu. Moreover the solution isC1,αcontinuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure
Asymptotic Behavior of Solutions of Systems of Neutral and Convolution Equations

AbstractSuppose J=[α,∞) for someα∈R or J=R and letXbe a Banach space. We study asymptotic behavior of solutions on J of neutral system of equations with values inX. This reduces to questions concerning the behavior of solutions of convolution equations (*)H∗Ω=b, whereH=(Hj,k) is anr×rmatrix,Hj,k∈D′L1,b=(bj) andbj∈D′(R,X), for 1⩽j,k⩽r. We prove that ifΩis a bounded uniformly continuous solution of (*) withbfrom some translation invariant suitably closed class A, thenΩbelongs to A, provided, for example, that detHhas countably many zeros on R andc0⊄X. In particular, ifbis (asymptotically) almost periodic, almost automorphic or recurrent,Ωis too. Our results extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhikov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (*). This leads to an extension of the known classes of almost periodicity to larger classes called mean-classes. We explore mean-classes and prove that bounded solutions of (*) belong to mean-classes provided certain conditions hold. These results seem new even for the simplest difference equationΩ(t+1)−Ω(t)=b(t) with J=X=R andbStepanoff almost periodic
On Moments of Negative Eigenvalues for the Pauli Operator

AbstractThis paper concerns the three-dimensional Pauli operatorP=(σ·(p−A(x)))2+V(x)with a non-homogeneous magnetic fieldB=curlA. The following Lieb–Thirring type inequality for the moment of negative eigenvalues is established,∑λj<0|λj|⩽C1∫R3|V(x)|5/2−dx+C2∫R3[bp(x)]3/2|V(x)|−dxwherep>3/2 andbp(x) is theLpaverage of |B| over a certain cube centered atxwith a side length scaling like |B|−1/2. We also show that, ifBhas a constant direction,∑λj<0|λj|γ⩽C1,γ∫R3|V(x)|γ+3/2−dx+C2,γ∫R3bp(x)|V(x)|γ+1/2−dxwhereγ>1/2 and p>1
Asymptotic of the Solutions of Hyperbolic Equations with a Skew-Symmetric Perturbation

AbstractUsing methods introduced by S. Schochet inJ. Differential Equations114(1994), 476–512, we compute the first term of an asymptotic expansion of the solutions of hyperbolic equations perturbated by a skew-symmetric linear operator. That result is first applied to two systems describing the motion of geophysic fluids: the rotating Euler equations and the primitive system of the quasigeostrophic equations. Finally in the last part, we study the slightly compressible Euler equations by application of that same result
Fluid-Dynamic Limit for the Centered Rarefaction Wave of the Broadwell Equation

AbstractWe study the asymptotic equivalence of the 1-d Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path. We consider the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the Euler equation contain centered rarefaction waves. In particular, Riemann data connected by rarefaction curves are included. We show that, as long as the initial data are a small perturbation of a non-vacuum constant state, the solution for the Broadwell equation exists globally in time and converges, in the small mean free path limit, to the solution of the corresponding Euler equation uniformly except for an initial layer whose width is essentially the order of the mean free path
Dirichlet Problem for the Diffusive Nicholson's Blowflies Equation

AbstractThe object of this paper is to consider the asymptotic behavior of solutions of the diffusive Nicholson's blowflies equation under Dirichlet boundary condition. A new approach is developed to study the global attractivity of the positive steady state for a reaction diffusion equation with time delay, when monotone or quasi–monotone conditions fail to apply. This approach should be applicable to other Dirichlet problems as well, even though our analysis is tailored to the diffusive Nicholson's blowflies equation
Bounds on the Number of Solutions for Elliptic Equations with Polynomial Nonlinearities

AbstractWe consider semilinear elliptic Neumann boundary value problems with polynomial nonlinearities. Suppose that the degreenof the polynomial is odd and that the coefficientanof the highest order term is strictly positive (such that the corresponding nonlinear operator is globally coercive); then, if the coefficients of the lower order terms are sufficiently small, the equation hasfor any given forcing termat mostnsolutions. The proof uses a Lyapunov–Schmidt procedure to reduce the problem to a one dimensional equation; using estimates on the lower order terms it is then shown that the one dimensional equation has at mostnsolutions
Traveling Waves, Riemann Problems and Computations of a Model of the Dynamics of Liquid/Vapor Phase Transitions

AbstractA model for the liquid/vapor phase transitions in a shock tube is discussed. Computations for the one-dimensional isothermal case is carried out to show that this model exhibits one-dimensional wave patterns observed in actual experiments. The existence of traveling waves under two different scalings are studied. For the first scaling, where the diffusion of different phases is very small relative to typical reaction time for the growth of phases while the viscosity is comparable to the reaction time, complete phase diagrams of traveling waves are obtained. Many of these traveling waves are undercompressive. Some of compressive traveling waves are unstable. For the second scaling, where viscosity, typical reaction time, and diffusion of different phases are comparable, the existence of traveling wave profiles of liquefaction and evaporation shocks when shock speeds are larger than some number is proved. These shocks are undercompressive. Evaporation shocks are rarefaction shocks. The nonexistence of these undercompressive shocks when the shock speed is smaller than some number is proved. It is observed through numerical computation that most of these traveling waves are unstable while some of them are stable or metastable. Riemann problems are considered. Admissibility of shocks involving phase changes under the second scaling is discussed. Solutions of Riemann problems with wave patterns observed in actual experiments are presented. Nonuniqueness of solutions of some Riemann problems is discussed
Characterization for the Kuratowski Limits of a Sequence of Sobolev Spaces

AbstractThe purpose of this paper is to study the strong lower and the weak upper limits in the sense of Kuratowski of a sequence of Sobolev spaces {W1,p0(Ωn)}n∈Nand compare them to a fixed spaceW1,p0(Ω). The results are expressed in terms of the behavior of the local capacity in balls of the complementary sets. If these two limits coeı&#x0308;ncide, we obtain necessary and sufficient conditions for the convergence in the sense of Mosco of the Sobolev spaces, hence for the continuity of the solution of a Dirichlet problem associated to thep-Laplacian in terms of the geometric domain variation
Weakly Nonlinear Limit for the Relaxation Approximation of Conservation Laws in Several Space Dimensions

AbstractIn this paper, the weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions is derived through asymptotic expansions and justified by employing the energy estimates. Compared with the work of G. Q. Chen, C. D. Levermore, and T. P. Liu (1994,Comm. Pure Appl. Math.47, 787–830), the main difficulty we confront in our analyzes lies in the fact that the problem of the existence of an (convex) entropy, which was essential in the proof of their estimate (5.29), for the system under our consideration is still an open problem since their Theorem 3.2 was proved only for the 2×2 case
Toda Soliton Limits on General Backgrounds

AbstractStarting from an arbitrary background solution of the Toda lattice, we study limits ofN-soliton solutions on this given background asNtends to infinity. This yields a new class of solutions of the Toda lattice. Simultaneously, we solve an inverse spectral problem for one dimensional Jacobi operators–we explicitly construct Jacobi operators whose spectrum contains a given (countable, bounded) set of eigenvalues and whose absolutely continuous spectrum coincides with that of a given background operator
The Fredholm Alternative at the First Eigenvalue for the One Dimensionalp-Laplacian

AbstractIn this work we study the range of the operatoru↦(|u′|p−2u′)′+λ1|u|p−2u,u(0)=u(T)=0,p>1. We prove that all functionsh∈C1[0,T] satisfying ∫T0h(t)sinp(πpt/T)dt=0 lie in the range, but that ifp≠2 andh≢0 the solution set is bounded. Here sin(πpt/T) is a first eigenfunction associated toλ1. We also show that in this case the associated energy functionalu↦(1/p)∫T0|u′|p−(λ1/p)∫T0|u|p+∫T0huis unbounded from below if 1<p<2 and bounded from below (with a global minimizer) ifp>2, onW1,p0(0,T) (λ1corresponds precisely to the best constant in theLp-Poincaré inequality). Moreover, we show that unlike the linear casep=2, forp≠2 the range contains a nonempty open set inL∞(0,T)
Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation

AbstractIn this paper, we consider a discrete delay problem with negative feedbackx(t)=f(x(t),x(t−1)) along with a certain family of time discretizations with stepsize 1/n. In the original problem, the attractor admits a nice Morse decomposition. We prove that the discretized problems have global attractors. It was proved by T. Gedeon and K. Mischaikov (1995,J. Dynamical Differential Equations7, 141–190) that such attractors also admit Morse decompositions. We then prove certain continuity results about the individual Morse sets, including that iff(x,y)=f(y), then the individual Morse sets are upper semicontinuous atn=∞
Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws

AbstractThe Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. We prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality
An Eigenfunction Expansion for a Quadratic Pencil of a Schrödinger Operator with Spectral Singularities

AbstractIn this paper, we consider the operatorLgenerated inL2(R+) by the differential expressionℓ(y)=−y″+[q(x)+2λp(x)−λ2]y,x∈R+=[0,∞),and the boundary conditiony(0)=0, wherepandqare complex-valued functions andpis continuously differentiable onR+. We derive a two-fold spectral expansion ofL(in the sense of Keldysh, 1951,Soviet Math. Dokl.77, 11–14 [1971,Russian Math. Survey26, 15–44 (Engl. transl.)]) in terms of the principal functions under the conditionslimx→∞p(x)=0,supx∈R+{eεx[|q(x)|+|p′(x)|]}<∞,ε>0,taking into account the spectral singularities. Also we investigate the convergence of the spectral expansion