pinecone/core-2020-05-10-deduplication huggingface dataset:
Vanishing Curvature Viscosity for Front Propagation

AbstractIn this paper we study the front propagation with constant speed and small curvature viscosity. We first investigate two related problems of conservation laws, one of which is on the nonlinear viscosity methods for the conservation laws, and the other one is on the structure of solutions to conservation laws with L1 initial data. We show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first-order rate of L1-convergence. The solutions of conservation laws with L1 initial data are shown to be bounded after t>0 if all singular points of initial data are from shocks. These results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed, as the small parameter goes to zero. After the front breaks down, the cusps will disappear promptly and corners will be formed
Stability in Semilinear Problems

AbstractIn the paper, we derive results concerning a continuous dependence of solutions on the right-hand side for a semilinear operator equation Lu=∇g(u), by assuming that L:D(L)⊂H→H (H−a Hilbert space) is self-adjont, with a closed range, and g:H→R is continuous convex on H and Gâteaux differentiable on D(L). Using these results, we obtain theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order ü+au=DuF(t, u, ω), t∈[0, π] a.e., with the Dirichlet boundary conditions u(0)=u(π)=0, where a⩾1, and ω is a functional parameter
Estimates for the Hill Operator, I

AbstractWe consider the Hill operator T=−d2/dt2+q(t) in L2(R), where q∈L2(0, 1) is a 1-periodic real potential. The spectrum of T consists of intervals σn=[λ−n−1, λ+n] separated by gaps γn=(λ−n, λ+n), n⩾1, with the lengths |γn|⩾0, and we assume λ+0=0. Let hn be a height of the corresponding slit in the quasimomentum domain and let ρn=π2 (2n−1)−|σn|>0 be the band shrinkage. We also have the gap gn, n⩾1, with the length |gn|, of the operator T⩾0. Introduce the sequences γ={|γn|}, h={hn}, g={|gn|}, ρ={ρn} and the norms ‖f‖2m=∑n⩾1(2πn)2mf2n, m⩾0. The following results are obtained: (i) double-sided estimates of ‖γ‖, ‖h‖1, ‖g‖1 in terms of ‖q‖2=∫10q(t)2dt, (ii) estimates of ‖ρ‖ in terms of ‖γ‖, ‖h‖1, ‖g‖1, ‖q‖, and (iii) a generalization of (i) and (ii) for more general potentials. The proof is based on the analysis of the quasimomentum as the conformal mapping, the embedding theorems and the identities
Necessary and Sufficient Conditions for Finite Time Singularities in Ordinary Differential Equations

AbstractThis paper gives a theorem which provides necessary and sufficient conditions for the existence of a real finite value of the independent variable at which the general solution of a certain class of ordinary differential equations diverges to infinity. This class is a large subset of the set of all autonomous, non-linear polynomial ordinary differential equation. These conditions involve the asymptotic form of the local series representation for the general solutions around the singularities and can be checked algorithmically
Singular Heteroclinic Cycles

AbstractWe study the unfolding of heteroclinic cycles of vector fields in Rn, that possess a hyperbolic singularity and a saddle-node. The principal eigenvalues at the hyperbolic singularity are assumed to be real, but the weak hyperbolic eigenvalues at the saddle-node may be either real or complex conjugate. We discuss the bifurcation diagrams of all codimension two such bifurcations. In some of the occurring cases chaotic dynamics appears in the unfolding. For the cases with only simple dynamics in the unfolding, we obtain the complete bifurcation diagram. We derive exponential expansions for the transition map near the saddle-node
Dynamics around Simple Resonant Tori in Nearly Integrable Hamiltonian Systems

AbstractBy an application of the K.A.M. theory, we derive an accurate normal form valid in the vicinity of partially hyperbolic tori which arise close to simple resonances in nearly integrable Hamiltonian systems. This normal form allows to detect orbits homoclinic to a persistent torus. Moreover, it also gives precise estimates on the times of transition around the stable and unstable manifolds of these tori. Hence, we provide an efficient tool to compute the speed of drift of orbits shadowing a chain of hyperbolic tori associated to a simple resonant curve in the action space
The Phase-plane Picture for a Class of Fourth-order Conservative Differential Equations

AbstractWe study the bounded solutions of a class of fourth-order equations −γu′′′′+u″+f(u)=0,γ>0. We show that when γ is not too large then the paths in the (u, u′)-plane of two bounded solutions do not cross. Moreover, the conserved quantity associated with the equation puts an ordering on the bounded solutions in the phase-plane and a continuation theorem shows that they fill up part of the phase-plane. We apply these results to the Extended Fisher–Kolmogorov (EFK) equation, a fourth-order model equation for bi-stable systems. The uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles the bounded solutions of the stationary EFK equation correspond exactly to those of the classical second-order Fisher–Kolmogorov equation. Besides, we establish the asymptotic stability of the heteroclinic solution of the EFK equation
Non-Existence Results for Semilinear Cooperative Elliptic Systems via Moving Spheres

AbstractBased upon a new development of the method of moving spheres, we introduce a new and general approach for non-existence of positive solutions of cooperative semilinear elliptic systems with the Laplacian as principal part. For supercritical nonlinearities we prove non-existence on bounded star-shaped domains. For subcritical nonlinearities we obtain non-existence results on a class of unbounded domains, which includes e.g. the entire space, certain curved halfspaces and the complement of bounded star-shaped domains. As a by-product we also get a symmetry result on halfspaces
On Super-high Frequencies in Discontinuous 1st-Order Delay–Differential Equations

AbstractOne considers a scalar 1st-order nonlinear differential equation with a delayed relay-output proportional feedback. One shows that, under a boundedness condition on the nonlinearity only, any solution of this equation has, after a finite time, a finite number of zeros on compact sets. An estimate of the time after which the super-high frequency disappears is provided. This improves some previous work by Shustin. As a consequence, using some work by Fridman et al., any solution of the system under study coincides, after a finite time, with one of the periodic solutions
Criteria for Cr Robust Permanence

AbstractLet xi=xifi(x) (i=1, …, n) be a Cr vector field that generates a dissipative flow φ on the positive cone of Rn. φ is called permanent if the boundary of the positive cone is repelling. φ is called Crrobustly permanent if φ remains permanent for sufficiently small Cr perturbations of the vector field. A necessary condition and a sufficient condition for Cr robust permanence involving the average per-capita growth rates ∫fidμ with respect to invariant measures μ are derived. The necessary condition requires that infμmaxi∫fidμ>0, where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone. The sufficient condition requires that the boundary flow admit a Morse decomposition {M1, …, Mk} such that every Mj satisfies minμmaxi∫fidμ>0 where the minimum is taken over invariant measures with support in Mj. As applications, we provide a sufficient condition for Cr robust permanence of Lotka–Volterra models and a topological characterization of Cr robust permanence for food chain models
The Method of Holomorphic Foliations in Planar Periodic Systems: The Case of Riccati Equations

AbstractWe develop a new method of investigation of periodic solutions of planar holomorphic polynomial differential equations with trigonometric coefficients. This method relies on analysis of geometry of leaves of certain holomorphic foliations in CP1×CP1. We concentrate our attention on the case of Riccati equations. We determine the dynamics of the non-autonomous equations, using some local invariants of holomorphic vector fields defining the foliations. In particular, new examples of Riccati equations without periodic solutions are presented
Nonlinear Stability of Strong Planar Rarefaction Waves for the Relaxation Approximation of Conservation Laws in Several Space Dimensions

AbstractIn this paper, we show that a strong planar rarefaction wave is nonlinear stable, namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions. Compared with former results obtained by T. P. Liu (1987, Comm. Math. Phys.108, 153–175) and T. Luo (1997, J. Differential Equations133, 255–279), our main novelty lies in the fact that the planar rarefaction waves do not need to be small, and in the one-dimensional case, the initial disturbance can also be chosen arbitrarily large
Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals Part II. Bifurcation of Periodic Orbits of Hamiltonian Systems

AbstractOur main results here are as follows: Let Xλ be a family of 2π-periodic Hamiltonian vectorfields that depend smoothly on a real parameter λ in [a, b] and has a known, trivial, branch sλ of 2π-periodic solutions. Let Pλ be the Poincaré map of the linearization of Xλ at sλ. If the Conley–Zehnder index of the path Pλ does not vanish, then any neighborhood of the trivial branch of periodic solutions contains 2π-periodic solutions not on the branch. Moreover, if each solution sλ is constant and each linearization Aλ of Xλ at sλ is time independent, then bifurcation of 2π-periodic orbits from the branch of equilibria arises whenever i(Ab)≠i(Ab), where i(A) is the index of the linear Hamiltonian system Ju=Au
Connecting Orbits from Synchronous Periodic Solutions to Phase-Locked Periodic Solutions in a Delay Differential System

AbstractWe consider a system of delay differential equations modelling the excitatory interaction of two identical neurons. Assuming the delay is sufficiently large, we show that the closure of the forward extension W5 of a 5-dimensional leading unstable manifold of the trivial solution contains a phase-locked periodic orbit and a synchronized periodic orbit and we classify the dynamics of the semiflow restricted to W5. We also obtain the precise information about the Floquet multipliers of the synchronized periodic orbit, which enables us to establish the existence of heteroclinic orbits from the synchronized periodic orbit to the phase-locked orbit
Some Classification Results for Hyperbolic Equations F(x, y, u, ux, uy, uxx, uxy, uyy)=0

AbstractWe provide a contact invariant characterization for equations of the formuxy+a(x, y, u)ux+b(x, y, u)uy+c(x, y, u)=0,uxy+a(x, y)ux+b(x, y)uy+c(x, y, u)=0,uxy+c(x, y, u)=0,uxy=0. We classify all equations of the form uxy+f(x, y, u, ux, uy)=0 for which the two Ovsiannikov's invariants are constants. These results include characterization of the Klein–Gordon equation uxy=u, the Liouville equation uxy=eu, and the class of Euler–Poisson–Darboux equations. It is shown that the wave equation uxy=0, Liouville equation, and the linear equation uxy=2u/(x+y)2 are the only variational equations Darboux integrable at level one. We also show that a hyperbolic Monge–Ampére equation Darboux integrable at level one is equivalent to an equation of type uxy+f(x, y, u, ux, uy)=0. We prove that the hyperbolic Fermi–Ulam–Pasta (FPU) equation uyy=κ(ux)2uxx is contact equivalent to a linear equation of type uxy=c(x+y)u and we classify all FPU equations Darboux integrable at level one. We also apply our results to equations of type uxy=F(u, ux) that describe pseudo-spherical surfaces
Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

AbstractIn the first part of this paper, it is proved that the number of limit cycles of the competitive three-dimensional Lotka–Volterra system in R3+ is finite if this system has not any heteroclinic polycycles in R3+. In the second part of this paper, a 3D competitive Lotka–Volterra system with two small parameters is discussed. This system always has a heteroclinic polycycle with three saddles. It is proved that there exists one parameter range in which the system is persistence and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistence and has at least one limit cycle
Finite Section Method for Linear Ordinary Differential Equations

AbstractSufficient conditions are given for the solution of a linear ordinary differential equation on the half line to be obtained as the limit of solutions of corresponding equations on finite intervals. Both the time-variant and the time-invariant case are considered, and in the latter case the sufficient conditions referred to above are also shown to be necessary. Included are applications to a differential equation that appears in linear transport theory, and to integral equations with semi-separable kernels
A General Approach to Stability in Free Boundary Problems

AbstractEvery solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem. The free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neumann data. We establish a rigorous stability analysis of such equilibria, including the construction of stable and unstable manifolds. For this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems. As an illustration we give two examples, the second being the focussing flame problem in combustion theory
Effect of Symmetry to the Structure of Positive Solutions in Nonlinear Eliptic Problems

AbstractWe consider the problem:Δu+hu+f(u)=0inΩRu=0on∂ΩRu>0inΩR,where ΩR≡{x∈RN∣R−1<|x|<R+1}, and the function f and the constant h satisfy suitable assumptions. This problem is invariant under the orthogonal coordinate transformations, in other words, O(N)-symmetric. We investigate how the symmetry affects to the structure of positive solutions. For a closed subgroup G of O(N), we consider a natural group action G×SN−1→SN−1. Then, we give a partial order on the space of G-orbits. Then, with respect to the partial order, a critical (locally minimal) orbital set will be defined. As a main result of this paper, we show that, when R→∞, a critical orbital set produces a solution of our problem whose energy is concentrated around a scaled critical orbital set
Small Data Blowup for Systems of Semilinear Wave Equations with Different Propagation Speeds in Three Space Dimensions

AbstractWe consider the Cauchy problem for a system of semilinear wave equations with small initial data and critical nonlinearity. As for a class of systems of quasilinear wave equations with critical nonlinearity, the small data global existence has been well developed for the case when the propagation speeds are distinct. In contrast with the quasilinear case, we show that the critical small data blowup occurs for the semilinear case, even if the propagation speeds are different from each other
Asymptotic Behavior of Solutions to a Hyperbolic System with Relaxation and Boundary Effect

AbstractIn this paper, we study the initial boundary value problem of the following hyperbolic system with relaxation[formula]on the half line R+ with the boundary conditions v(0, t)=v−. When the asymptotic states are stationary wave or rarefaction wave or superposition of these two kind waves, we prove the stability of these wave patterns for small perturbation. The study is motivated by [7] where the asymptotic behavior of solutions to the scalar viscous conservation law with boundary corresponding to rarefaction waves was studied. In our analysis, we do not require (v+, u+) satisfy the equilibrium equation, i.e., u+=f(v+) as in [15]
Gevrey Regularity for the Attractor of a Partially Dissipative Model of Bénard Convection in a Porous Medium

AbstractConvective flow though a porous medium can be modeled by Darcy's law—a linear, weakly damped momentum equation—coupled with an advection–diffusion equation for the energy. The solution semigroup for this system is not smoothing, and the solution of the momentum equation does not gain regularity with respect to its initial value in finite time. However, it is known that the semigroup is asymptotically smoothing, so that the system possesses a finite dimensional global attractor as well as exponential attractors. We show that the global attractor is contained in a special Gevrey class of regularity and, in particular, is real analytic. The key idea is the use of a Fourier splitting method to approximate every orbit asymptotically in time by a Gevrey-regular function
Uniform Anti-maximum Principles

AbstractConsider a second or higher order elliptic partial differential equation Au=λu+f on an open bounded domain Ω of Rn with homogeneous boundary conditions Bu=0. If there exists a simple eigenvalue for which the corresponding eigenfunction is positive and satisfies appropriate boundary estimates, then an anti-maximum principle holds. For positive f∈Lp(Ω) with p large enough there exists δf>0 such that for λ∈(λ1, λ1+δf) the solution is negative and for λ∈(λ1−δf, λ1) the solution is positive. We give conditions such that this sign reversing property is uniform: there is δ>0 such that for all positive f the solution u is negative for λ∈(λ1, λ1+δ) and positive for λ∈(λ1−δ, λ1). Two classes of higher order boundary value problems that satisfy these conditions will be given
Oscillation Criteria for Linear Matrix Hamiltonian Systems

AbstractIn this paper we present oscillation criteria in terms of the coefficient functions for the matrix linear Hamiltonian system U′=A(x)U+B(x)V, V′=C(x)U−A*(x)V under the hypothesis H:A(x), B(x)=B*(x)>0 and C(x)=C*(x) are n×n matrices of real valued continuous functions on the interval I=[a, ∞), (−∞<a). These criteria generalize earlier results due to Etgen and Pawlowski (1976), Erbe, Qingkai Kong and Shigui Ruan (1993), and Fanwei Meng, Jizhong Wang, and Zhaowen Zheng (1998) for the system (P(x)U′)′+Q(x)U=0 or its special cases
Uniqueness of the Stationary Wave for the Extended Fisher–Kolmogorov Equation

AbstractThe extended Fisher–Kolmogorov equation, ut=−βuxxxx+uxx+u−u3, β>0, models a binary system near the Lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria ±1. For the classical case, β=0, the heteroclinic is u(x)=tanh(x/2) and is unique up to the obvious symmetries. We prove the conjecture that the uniqueness persists all the way to β=1/8, where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur. Our methods are non-perturbative and employ a global cross-section to the Hamiltonian flow of the stationary fourth order equation on the energy level of ±1. We also prove uniform a priori bounds on all bounded stationary solutions, valid for any β>0
Singular Point-like Perturbations of the Bessel Operator in a Pontryagin Space

AbstractThe spectral problem for the Bessel equation of order ν on (0, ∞) in the case 0<ν<1 is closely related to the Nevanlinna function Q(z)=−π(−z)ν/(2sinπν). If ν>1 and ν≠2, 3, ..., this function belongs to the generalized Nevanlinna class Nm, m=[ν+12]. A natural question appears: To what spectral problem does this function correspond? We answer this and related questions using Pontryagin space operator realizations of suitable singular point-like perturbations of the Bessel operator. In this paper we discuss the spectra of these realizations and we derive eigenfunction expansions via related wave operators
Evolution Semigroups and Sums of Commuting Operators: A New Approach to the Admissibility Theory of Function Spaces

AbstractThis paper is concerned with conditions for the admissibility of a translation invariant function space M with respect to a well posed linear evolution equation du/dt=Au+f(t), t∈R (*). We propose a new approach to this problem by considering the sum of two commuting operators −d/dt≔−DM and the operator of multiplication by A on M. On the one hand, the closure of this operator is the infinitesimal generator of the so-called evolution semigroup associated with (*). On the other hand, the generator G of this semigroup relates a mild solution u of (*) to the forcing term f by the rule Gu=−f. Consequently, various spectral criteria of the type σ(DM)∩σ(A)=∅ for the admissibility of the function space M with respect to (*) can be proved in an elegant manner. Moreover, they can be naturally extended to general classes of differential equations, including higher order and abstract functional differential equations. Applications and examples are provided to illustrate the obtained results
Existence and Non-Existence of Global Smooth Solutions for p-System with Relaxation

AbstractIn this paper, we consider the Cauchy problem for p-system with relaxation. Under the assumption that the relaxation time ε is sufficiently small, we prove the existence of the global smooth solution to the Cauchy problem with C1-initial data provided the C0-norm of the derivative of the initial data is of the order of ξ/ε. Here ξ is a small positive constant. On the other hand, when the initial density has compact support but is not identically zero, we prove the global regular solution for the Cauchy problem does not exist
Filippov's and Filippov–Ważewski's Theorems on Closed Domains

AbstractThe celebrated Filippov's theorem implies that, given a trajectory x1:[0, +∞[↦Rn of a differential inclusion x′∈F(t, x) with the set-valued map F measurable in t and k-Lipschitz in x, for any initial condition x2(0)∈Rn, there exists a trajectory x2(·) starting from x2(0) such that |x1(t)−x2(t)|⩽ekt|x1(0)−x2(0)|. Filippov– Ważewski's theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x′∈coF(t, x) by a trajectory of the original inclusion x′∈F(t, x) starting from the same initial condition. In the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset Θ⊂Rn. The latter is allowed to be non smooth. We impose a generalized Soner type condition on F and Θ, yielding extensions of the above classical results to infinite horizon constrained problems. Applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well
Ljusternik–Schnirelmann Theorem for the Generalized Laplacian

AbstractLet m:[0, ∞[→[0, ∞[ be an increasing continuous function with m(t)=0 if and only if t=0, m(t)→∞ as t→∞ and Ω⊂RN a bounded domain. In this paper we show that for every r>0 the problem [formula]has an infinite number of eigenfunctions on the level set ∫ΩM(|∇u|)=r, where M(t)=∫|t|0m(s)ds and g:R→R is odd satisfying some growth condition. Moreover, we show that the sequence of associated eigenvalues tends to infinity. We emphasize that no ▵2-condition is needed for M or for its conjugate, so the associated functionals are not continuously differentiable, in general
Energy Decay Rate of Wave Equations with Indefinite Damping

AbstractWe consider the one-dimensional wave equation with an indefinite sign damping and a zero order potential term. Using a shooting method, we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients. In addition, if the damping coefficient is “more positive than negative,” we prove that the energy of system decays uniformly exponentially to zero. This sharp result generalizes a previous work of Freitas and Zuazua (1996)