pinecone/core-2020-05-10-deduplication huggingface dataset:
Reformulations of Measure Differential Inclusions and Their Closed Graph Property

AbstractNew formulations are given for measure differential inclusions, “dx/dt∈K(t)⊂Rn,” where x(·) is a function of bounded variation and K is a set-valued map with closed convex values and has closed graph. Measure differential inclusions were first named by J. J. Moreau for studying rigid body with impacts, shocks and Coulomb friction and assumed that K(t) is always a cone. The new formulations are used to show that the graph of the solution operator is closed under point- wise convergence of the solutions x(·) and weak* convergence of the differential measures dx, provided that the minimum norm points of K(t) are bounded and the asymptotic cones K(t)∞ are always pointed
Long Range Scattering and Modified Wave Operators for some Hartree Type Equations, III. Gevrey Spaces and Low Dimensions

AbstractWe study the theory of scattering for a class of Hartree type equations with long range interactions in arbitrary space dimension n⩾1, including the case of Hartree equations with time dependent potential V(t, x)=κtμ−γ|x|−μ with 0<γ⩽1 and 0<μ<n. This includes the case of potential V(x)=κ|x|−γ and can be extended to the limiting case of nonlinear Schrödinger equations with cubic nonlinearity κtn−γ|u|2u. Using Gevrey spaces of asymptotic states and solutions, we prove the existence of modified local wave operators at infinity with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators, thereby extending the results of previous papers which covered the range 0<γ⩽1 but only 0<μ⩽n−2 and were therefore restricted to space dimension n⩾3
Averaging the Planar Three-Body Problem in the Neighborhood of Double Inner Collisions

AbstractLevi–Civita's regularization procedure for the two-body problem easily extends to a regularization of double inner collisions in the system consisting of two uncoupled Newtonian two-body problems. Some action-angle variables are found for this regularization, and the inner body is shown to describe ellipses on all energy levels. This allows us to define a second projection of the phase space onto the space of pairs of ellipses with fixed foci. It turns out that the initial and regularized averaged Hamiltonians of the three-body problem agree, when seen as functions on the space of pairs of ellipses. After the reduction of the problem by the symmetry of rotations, the initial and regularized averaged planar three-body problems are shown to be orbitally conjugate, up to a diffeomorphism in the parameter space consisting of the masses, the semi major axes and the angular momentum
Existence of Pseudo-Conformally Invariant Solutions to the Davey–Stewartson System

AbstractWe prove that the pseudo-conformal group of the Schrödinger equation acts on solutions of the Davey–Stewartson system and that there exists an infinite family of solutions which are invariant under the action of that group. We also exhibit two different time behaviors for these invariant solutions of the Davey–Stewartson system, and we study the stability of some of these solutions, and prove that initial data close to them give rise to global solutions asymptotically behaving like an invariant solution
Perturbations from an Elliptic Hamiltonian of Degree Four I. Saddle Loop and Two Saddle Cycle

AbstractThe paper deals with Liénard equations of the form x=y, y=P(x)+yQ(x) with P and Q polynomials of degree respectively 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree 4 and especially to the study of the related elliptic integrals. Besides some general results the paper contains a complete treatment of the Saddle Loop case and the Two Saddle Cycle case. It is proven that the related elliptic integrals have at most two zeros, respectively one zero, the multiplicity taken into account. The bifurcation diagram of the zeros is also obtained
Perturbations from an Elliptic Hamiltonian of Degree Four II. Cuspidal Loop

AbstractThe paper deals with Liénard equations of the form x=y, y=P(x)+yQ(x) with P and Q polynomials of degree respectively 3 and 2. Attention goes to perturbations of the Hamiltonian vector field with an elliptic Hamiltonian of degree 4, exhibiting a cuspidal loop. It is proven that the least upper bound for the number of zeros of the related elliptic integral is four, and this upper bound is a sharp one. This permits to prove the existence of Liénard equations of type (3, 2) with at least four limit cycles. The paper also contains a complete result on the respective number of “small” and “large” limit cycles
Multiple Bifurcations in a Delayed Predator–Prey System with Nonmonotonic Functional Response

AbstractA delayed predator–prey system with nonmonotonic functional response is studied by using the normal form theory of retarded functional differential equations developed by Faria and Magalhães. The bifurcation analysis of the model indicates that there is a Bogdanov–Takens singularity for any time delay value. A versal unfolding of the model at the Bogdanov–Takens singularity is obtained. On the other hand, it is shown that small delay changes the stability of the equilibrium of the model for some parameters and the system can exhibit Hopf bifurcation as the time delay passes through some critical values
Weighted Decay Estimates for the Wave Equation

AbstractIn this work we study weighted Sobolev spaces in Rn generated by the Lie algebra of vector fields (1+|x|2)1/2∂xj,j=1, …, n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rn. As an application we derive weighted Lq estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math.119, 1291–1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces
Uniform Bounds for Solutions to Quasilinear Parabolic Equations

AbstractWe consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p-Laplacian operator, u=Δpu≔∑di=1∂i(|∇u|p−2∂iu) with p∈[2, d), on a domain D⊂Rd of finite measure. We prove that |u(t, x)|⩽c|D|αt−β‖u0‖γr for all t>0,x∈D and for all initial data u0∈Lr(D), provided r is not smaller than a suitable r0, where α, β, γ are positive constants explicitly computed in terms of d, p, r. The nonlinear cases associated with the case p=2 display exactly the same contractivity properties which hold for the linear heat equation. We also show that the nonlinear evolution considered is contractive on any Lq space for any q∈[2, +∞]
Lyapunov's Second Method for Random Dynamical Systems

AbstractThe method of Lyapunov functions (Lyapunov's second or direct method) was originally developed for studying the stability of a fixed point of an autonomous or non-autonomous differential equation. It was then extended from fixed points to sets, from differential equations to dynamical systems and to stochastic differential equations. We go one step further and develop Lyapunov's second method for random dynamical systems and random sets, together with matching notions of attraction and stability. As a consequence, Lyapunov functions will also be random. Our test is that the extension be coherent in the sense that it reduces to the deterministic theory in case the noise is absent, and that we can prove that a random set is asymptotically stable if and only if it has a Lyapunov function. Several examples are treated, including the stochastic Lorenz system
Existence of Nonnegative and Nonpositive Solutions for Second Order Periodic Boundary Value Problems

AbstractWe are interested in nonnegative and nonpositive solutions of the boundary value problem u″=f(t, u), u(0)=u(1), u′(0)=u′(1), where f fulfils the Carathéodory conditions on [0, 1]×R. We generalize the results reached by M. N. Nkashama, J. Santanilla and L. Sanchez and present estimates for solutions. In addition, we apply our existence theorems to periodic boundary value problems for nonlinear Duffing equations whose right-hand sides have a repulsive or attractive singularity at the origin. We extend or generalize existence results by A. C. Lazer and S. Solimini and other authors. Moreover, we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity, in constrast to the previous papers on this subject. Our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples
Regularity of the Interfaces with Sign Changes of Solutions of the One-Dimensional Porous Medium Equation

AbstractIn previous papers we considered the Cauchy problem for the one-dimensional evolution p-Laplacian equation for nonzero, bounded, and nonnegative initial data having compact support, and showed that after a finite time the set of spatial critical points of the nonnegative solution u=u(x, t) in {u>0} consists of one point, the spatial maximum point of u, and the curve of the spatial maximum points is continuous with respect to the time variable. Since the spatial derivative ∂xu satisfies the porous medium equation with sign changes, the curve of the spatial maximum points is regarded as an interface with sign changes of ∂xu. On the other hand, in a paper by M. Bertsch and D. Hilhorst (1991, Appl. Anal.41, 111–130) the interfaces where the solutions change their sign were studied in detail for the initial-boundary value problems of the generalized porous medium equation over two-dimensional cylinders. But the monotonicity of the initial data is assumed there. As is noted in Section 4 of our earlier work (1996, J. Math. Anal. Appl.203, 78–103), the monotonicity of ∂xu(ċ, t) in some neighborhood of the spatial maximum point of u(ċ, t) cannot be assumed, and therefore, if this monotonicity for some large t>0 is proved, then by the method of Bertsch and Hilhorst (cited above) one may get more precise regularity properties of the curve of the spatial maximum points. The purpose of the present paper is twofold. One is to remove some monotonicity assumption for initial data in Bertsch and Hilhorst's theorem concerning the regularity of the interfaces with sign changes of solutions of the one-dimensional generalized porous medium equation. By comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time. The other is as a by-product of the method to get C1 regularity of the curves of the spatial maximum points of nonnegative solutions of the Cauchy problem for the evolution p-Laplacian equation for sufficiently large t
Properties of the Principal Eigenvalues of a General Class of Non-classical Mixed Boundary Value Problems

AbstractIn this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions. Our theory is a substantial extension of the classical theory by P. Hess and T. Kato (1980, Comm. Partial Differential Equations5, 999–1030). In obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself. These auxiliary results complement and in some sense complete the theory of D. Daners and E. N. Dancer (1997, J. Differential Equations138, 86–132). The main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to H. Amann and J. López-Gómez (1998, J. Differential Equations146, 336–374)
Small Periodic Solutions Generated by Sublinear Terms

AbstractThe paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained
Quasistatic Motion of a Capillary Drop I. The Two-Dimensional Case

AbstractA theory is presented for analyzing the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia. The theory is developed here on the two-dimensional version of the relevant free-boundary model for Stokes equations. As we show, the two-dimensional problem presents most of the difficulties expected from a projected three-dimensional study while allowing for simpler manipulation of the spherical harmonics. Within this context we show that if the free-boundary initiates close to a circler=1+ελ0(θ),|ε|small, then there exists a global-in-time solution with free boundaryr=1+λ(θ, t, ε)=1+∑n⩾1λn(θ, t)εn, which approaches a circle exponentially fast as t→∞. Moreover, we prove that if λ0(θ) is analytic (resp. C∞) in θ, then the velocity u(x, t, ε), the pressure p(x, t, ε), and the free boundary λ are all jointly analytic (resp. C∞) in (x, ε)
Diffusive–Dispersive Traveling Waves and Kinetic Relations Part I: Nonconvex Hyperbolic Conservation Laws

AbstractMotivated by the theory of phase transition dynamics, we consider one-dimensional, nonlinear hyperbolic conservation laws with nonconvex flux-function containing vanishing nonlinear diffusive–dispersive terms. Searching for traveling wave solutions, we establish general results of existence, uniqueness, monotonicity, and asymptotic behavior. In particular, we investigate the properties of the traveling waves in the limits of dominant diffusion, dominant dispersion, and asymptotically small or large shock strength. As the diffusion and dispersion parameters tend to 0, the traveling waves converge to shock wave solutions of the conservation law, which either satisfy the classical Oleinik entropy criterion or are nonclassical undercompressive shocks violating it
Structure of Positive Radial Solutions to Scalar Field Equations with Harmonic Potential

AbstractWe study a nonlinear elliptic equation with a harmonic potential−Δ u+(λ+|x|2)u−|u|p−1u=0,x∈Rn, (E) which is related to standing waves for a nonlinear Schrödinger equation. The existence of positive solution for (E) in some energy space can be shown by the standard variational method under n⩾1, λ>−n and 1<p<(n+2)/(n−2)+. However, the uniqueness is a delicate problem. In this paper, noting that all positive solutions of (E) in the energy space are radially symmetric about the origin, we study the structure of positive radial solutions to (E) and show the uniqueness of positive solution with finite energy when n⩾3
Variational Problems for a Class of Functionals on Convex Domains

AbstractLet Ω be a bounded convex open subset of RN, N⩾2, and let J be the integral functionalJ(u)≐∫Ω[f(|Du(x)|)−u(x)]dx, where f:[0, +∞[→R∪{+∞} is a lower semicontinuous function (possibly nonconvex and with linear growth). We prove that the functional J admits a unique minimizer in the space of W1, 10(Ω) functions that depend only on the distance from the boundary of Ω, provided that the ratio between the Lebesgue measure of Ω and the (N−1)-dimensional Hausdorff measure of ∂Ω is strictly less than a constant related to the growth of f at infinity
A Wavefront Tracking Algorithm for N×N Nongenuinely Nonlinear Conservation Laws

AbstractWe introduce a wavefront tracking algorithm for N×N hyperbolic systems of conservation lawsut+F(u)x=0, that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue λi(u) of DF(u) in the direction of the ith right eigenvector ri(u), vanishes on a single (N−1)-dimensional hypersurface in the u-space, transversal to the field ri(u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc.30 (1981), no. 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S.A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax
Asymptotic Behaviour for a Phase-Field System with Hysteresis

AbstractThe method of hysteresis operators in modelling phase transitions is applied here to the problem of asymptotic stabilization of solutions to a phase-field system with hysteresis. While it is known that a unique global strong solution exists for every initial data and that the evolution process described by this system is thermo-dynamically consistent in the sense that the absolute temperature remains positive for all times and the Clausius–Duhem inequality holds almost everywhere, we study here the asymptotic behaviour of the solution as t→∞
Singular Sturm–Liouville Theory on Manifolds

AbstractIn this paper we investigate Schrödinger operators L=−Δg+a(x) on a compact Riemannian manifold (M, g), where the potential function a(x) is assumed to be continuous, but not necessarily bounded, outside of some closed set Σ⊂M of measure zero. Under certain geometric hypotheses on Σ and growth conditions on a(x) as x→Σ, we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a(x) is allowed to approach −∞ as x→Σ. We also consider conditions on Σ and a(x) under which the Sturm–Liouville theory of L is “singular” in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially self-adjoint or more generally “essentially Dirichlet” (a new property that we define). The behavior of L on weighted Sobolev spaces is also discussed. In most of the paper we assume that Σ is a k-dimensional submanifold without boundary, but in the last few sections we generalize our results to stratified sets
The General Properties of Discrete-Time Competitive Dynamical Systems

AbstractFor strongly competitive discrete-time dynamical systems on a strongly ordered topological vector space, it is proved that any α- or ω-limit set is unordered and lies on some invariant hypersurface with codimension one, which generalizes Hirsch's results for competitive autonomous systems of ordinary differential equations to competitive maps in a very general framework and implies the Sarkovskii's Theorem for planar strongly competitive and monotone maps. We also show that the definition of competitiveness gives more restriction on homeomorphic property
Left-Definite Sturm–Liouville Problems

AbstractLeft-definite regular self-adjoint Sturm–Liouville problems, with either separated or coupled boundary conditions, are studied. We give an elementary proof of the existence of eigenvalues for these problems. For any fixed equation, we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions. Some of our results here yield an algorithm for numerically computing the eigenvalues of a left-definite problem with an arbitrary coupled boundary condition. Our inequalities imply that the well-known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case. Moreover, we study the continuous and differentiable dependence of the eigenvalues of the general left-definite problem on all the parameters in its differential equation and boundary condition
Isolated Singularities for Fully Nonlinear Elliptic Equations

AbstractWe obtain Serrin type characterization of isolated singularities for solutions of fully nonlinear uniformly elliptic equations F(D2u)=0. The main result states that any solution to the equation in the punctured ball bounded from one side is either extendable to the solution in the entire ball or can be controlled near the centre of the ball by means of special fundamental solutions. In comparison with semi- and quasilinear results the proofs use the viscosity notion of generalised solution rather than distributional or Sobolev weak solutions. We also discuss one way of defining the expression −P+λ, Λ(D2u), (P−λ, Λ(D2u)) as a measure for viscosity supersolutions (subsolutions) of the corresponding equation. Here P±λ, Λ are the Pucci extremal operators
Reversible Relative Periodic Orbits

AbstractWe study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along group orbits. Our results extend those recently obtained in the equivariant context by B. Sandstede et al. (1999, J. Nonlinear Sci.9, 439–478) and C. Wulff et al. (2001, Ergodic Theory Dynam. Systems21, 605–635)
The Brézis–Nirenberg Problem on S3

AbstractIn this paper we study existence and nonexistence of solutions to the Brézis–Nirenberg problem for different values of λ in geodesic spheres on S3. The picture differs considerably from the one in the Euclidean space. It is shown that large spheres containing the hemisphere have two different type of radial solutions for negative values of λ. Numerical results indicate that for λ very small the solutions have a maximum near the boundary, whereas for larger values of λ the maximum is at the origin. The techniques used are: Pohozaev type identities, concentration-compactness lemma and numerical methods
Nonlinear Geometric Optics for Short Pulses

AbstractThis paper studies the propagation of pulse-like solutions of semilinear hyperbolic equations in the limit of short wavelength. The pulses are located at a wavefront Σ≔{φ=0} where φ satisfies the eikonal equation and dφ lies on a regular sheet of the characteristic variety. The approximate solutions are uεapprox=U (t, x, φ(t, x)/ε) where U(t, x, r) is a smooth function with compact support in r. When U satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions uεexact such that uεapprox has relative error O(ε) as ε→0. While the transport equation is familiar, the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope
Finite Cyclicity of Graphics with a Nilpotent Singularity of Saddle or Elliptic Type

AbstractIn this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside C∞ families of planar vector fields. In some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3. By blowing up the family of vector fields, we obtain all the limit periodic sets. We calculate two different types of Dulac maps in the blown-up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point. The finite cyclicity theorems are derived by a generalized derivation–division method introduced by Roussarie