pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Scaling Limits at Zeros of Solutions of Parabolic Equations

AbstractWe are concerned with the local scaling limits near zero points of solutions of general parabolic equations. We show that at each zero point of finite vanishing order, the solution is asymptotic to a homogeneous polynomial solution of the “osculating” equation with constant coefficients. This provides useful information about the local structure of zero sets of solutions. In particular, we give upper bounds for Hausdorff dimensions of zero sets of weak solutions. Our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems
Analyticity and Differentiability of Semigroups Associated with Elastic Systems with Damping and Gyroscopic Forces

AbstractIn this paper, we consider a second order variational evolution equation in a Hilbert space, which can model an elastic system with damping and gyroscopic forces. A new form of the corresponding first order evolution equation is introduced, and its well-posedness is proved by means of the semigroup theory. We give sufficient conditions for analyticity and differentiablity of the associated semigroup. The results are applied to several PDEs with discontinuous coefficients and mechanical boundary conditions. It is proved that Russell's spacial hysteresis model of a vibrating beam [22] is associated with an exponentially stable, analytic semigroup
Vortex-Condensations of a Non-Relativistic Maxwell–Chern–Simons Theory

AbstractWe study an elliptic system subject to periodic boundary conditions arising from the study of vortex-condensates for a selfdual non-relativistic model introduced by Dunneet al., Phys. Rev. D.43, 1332–1345 (1990) in the framework of Chern–Simons gauge theory. We prove the existence of a vortex-condensate forallvalues of the relevant physical parameters involved in the model. This completes a recent work of Spruck and Yang [16] who could deal only with a restricted set of those parameters. Uniqueness is also discussed
Inverse Problems and Chaotic Dynamics of Parabolic Equations on Arbitrary Spatial Domains

AbstractLetΩ⊂RNbe an arbitrary smooth bounded domain. We prove the existence of a polynomial functiona(x) on RNsuch that an arbitrary (and sufficiently small) vector field on RN+1can be realized on the center manifold of the semilinear parabolic equation[formula]with an appropriate nonlinearityg: (x,s,w)∈Ω×R×RN↦g(x,s,w)∈R. This extends earlier results of Poláčik and Rybakowski and shows that arbitrary chaotic behavior is possible for semilinear scalar parabolic BVPs on arbitrary bounded domainsΩ. To prove this we establish some spectral perturbation and convergence results and solve an abstract inverse eigenvalue problem
Center Manifolds for Infinite Dimensional Nonautonomous Differential Equations

AbstractWe study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions. We assume that the linear part of the equation admits, in a very general sense, a decomposition into center and hyperbolic parts. The center manifold is obtained directly as the graph of a fixed point for a Lyapunov–Perron type integral operator. We prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operatorΛ. The operatorΛis formed by the Green's function for the hyperbolic part and composition operators induced by a flow on the center part. We formulate the usual gap condition in spectral terms and show that this condition is, in fact, a condition of boundedness ofΛon corresponding spaces of differentiable functions. This gives a direct proof of the existence of a smooth global center manifold
Weak Solution to Some Penrose–Fife Phase-Field Systems with Temperature-Dependent Memory

AbstractIn this paper a phase-field model of Penrose–Fife type is considered for a diffusive phase transition in a material in which the heat flux is a superposition of two different contributions: one part is proportional to the spatial gradient of the inverse temperature, while the other is of the form of the Gurtin–Pipkin law introduced in the theory of materials with thermal memory. It is shown that an initial-boundary value problem for the resulting state equations has a unique solution, thereby generalizing a number of recent results
Global Bifurcation Results for Semilinear Elliptic Equations on RN: The Fredholm Case

AbstractWe prove the existence of a continuum of positive solutions for the semilinear elliptic equation −Δu(x)=λg(x)f(u(x)), 0<u<1 forx∈RN, lim|x|→+∞u(x)=0, which arises in population genetics, under the hypotheses thatN⩾3 and the weightgchanges sign, being negative and away from zero at ∞. After establishing the existence of a simple positive principal eigenvalueλ1for the corresponding linearized problem, we prove the existence of a continuum of solutions lying in the space R×H2extended fromλ1to ∞. To complete this task we state a new version of the global bifurcation theory for nonlinear Fredholm (noncompact) operators and prove the compactness of the solution set of the problem
Boundedness of Solutions of Nonlinear Differential Equations

AbstractIn this paper we are concerned with the boundedness of solutions of some nonlinear differential equations, which are neither dissipative nor conservative, having the formx″+(x, t)x′+g(x, t)=0, wherefandgare odd polynomials inxwith coefficients which are even and periodic int. Using the KAM theory for reversible systems, we prove that all the solutions are bounded whenever a sharp condition on the degrees offandgis satisfied. We also obtain a boundedness result whenf(x, t)≡0 (i.e., the equation is conservative), under some smoothness assumptions ongwhich improve the previously known ones. In this case, no symmetry conditions ongare needed
A Nonlinear Parabolic Problem in an Exterior Domain

AbstractWe consider the problem:[formula]Herep>1,N⩾2, 0 is a finite union of disjoint open sets, anduo(x) is a continuous, nonnegative, and bounded function such thatuo(x)∼A |x|−αas|x|→∞, ((0.4))for someA>0 andα>0. In this paper we discuss the asymptotic behaviour of solutions to (0.1)–(0.4) in terms of the various values of the parametersp,A,N,Ω, andα. A common pattern that emerges from our analysis is the existence of an external zone whereu(x,t)∼uo(x), and one or several internal regions, where the influence of the setΩ, as well as that of diffusion and absorption, is most strongly felt. We present a complete classification of the stabilization profiles in terms of these parameters
A Principle of Reduced Stability for Reaction–Diffusion Equations

AbstractA semilinear elliptic boundary value problem,Au+f(x,u,λ)=0 (withfu(x,u,λ) bounded below) can be shown to be equivalent to a finite-dimensional problem,B(c,λ)=0∈Rd(c∈Rd), in the sense that their solution sets, which are not necessarily singletons, are in a one–to–one correspondence (c(u)↔u(c)). The functionB(c,λ) is called the bifurcation function. It is shown that, for any solutionu(c), the number of negative (resp. zero) eigenvalues of the matrixBc(c,λ) is identical to the number of negative (resp. zero) eigenvalues of the linearized elliptic operatorAv+fu(x,u(c),λ)v. This results in a version of the principle of reduced stability for the problemut+Au+f(x,u,λ)=0 and its reductionc′+B(c,λ)=0
Boundary Layers for Viscous Perturbations of Noncharacteristic Quasilinear Hyperbolic Problems

AbstractIn this paper, we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero. The boundary is noncharacteristic for the hyperbolic system. We in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions. A counterexample is given when this condition is not satisfied, and the solution blows up as the viscosity goes to 0
Asymptotic Unfoldings of Dynamical Systems by Normalizing beyond the Normal Form

AbstractSystematic methods are given for simplification of the normal form of a perturbed dynamical system near a rest point. It then becomes easy to write down an unfolding of the dynamical system which can be rigorously justified as a universal unfolding with respect to asymptotic equivalence. These unfoldings always have finite codimension. They are both weaker and stronger than topological unfoldings: they may fail to be universal unfoldings in the topological sense because of behavior which escapes detection by asymptotic methods, and they also may distinguish asymptotic behaviors that are topologically equivalent. The method is illustrated by computing an unfolding of codimension 14, as well as some standard examples of codimension 2 and 3
Formation of Singularities in Compressible Euler–Poisson Fluids with Heat Diffusion and Damping Relaxation

AbstractWe are concerned with a coupling effect of heat diffusion, damping relaxation, and electric potential on the large-time behavior of solutions of the compressible Euler–Poisson fluids starting with smooth initial data. In this paper we prove that, for smooth initial data with large C1norm, the corresponding solution will develop singularities in finite time. This fact indicates that this coupling effect is not strong enough to prevent the formation of singularities and the development of shock waves in compressible Euler–Poisson fluids as time evolves
Abstract Forced Symmetry Breaking and Forced Frequency Locking of Modulated Waves

AbstractWe consider abstract forced symmetry breaking problems of the typeF(x, λ)=y. It is supposed that for allλthe mapsF(·, λ) are equivariant with respect to the action of a compact Lie group, thatF(x0, λ0)=0 and, hence, thatF(x, λ0)=0 for all elementsxof the group orbit O(x0) ofx0. We look for solutionsxwhich bifurcate from the solution family O(x0) asλandymove away fromλ0and zero, respectively. Especially, we describe the number of different solutionsx(for fixed control parametersλandy), their dynamic stability and their asymptotic behavior forytending to zero. Further, generalizations are given to problems of the typeF(x, λ)=y(x, λ). Finally, our results are applied to a forced frequency locking problem of the typex(t)=f(x(t), λ)−y(t). Here it is supposed that the vector fieldsf(·, λ) areS1-equivariant, that the unperturbed equationx=f(x, λ0) has an orbitally stable modulated wave solution and that the forcingy(t) is a modulated wave
Asymptotic Behavior of Some Linear Differential Systems

AbstractConsider the following chasing game in R amongbparticles whose trajectories are governed by the following differential equations: Fori∈S={1, 2, …, b},F′i(t)=∑j≠iqij(t)(Fj(t)−Fi(t)),whereFi(t) is the position at timetof the particleiandqij(t), which measures the instantaneous rate that particleiis attracted toward particlej, is given byqij(t)=pijλ(t)U(i, j)forj≠i. The rate functionλ(t) is positive with limt→∞λ(t)=0,U(i, j)∈[0, ∞] is the “cost” function indicating the level of difficulty fromitojand the nonnegative numberspijfori≠jreflect the neighborhood structure of the particles. Particles are attracted to one another but such attraction becomes weaker as time goes on. A sufficient condition onλ(t) is given for allFi(t) converging to a common finite limit
Singular Internal Stabilization of the Wave Equation

AbstractWe consider an initial and boundary value problem for the one and two dimensional wave equation with nonlinear damping concentrated on an interior point and respectively on an interior curve. In the two dimensional case our main result asserts that generically (i.e., for almost all interior curves) the solutions decay to zero in the energy space. When the domain is strictly convex we show that, whatever the interior curve is, the decay is not uniform. We generalize in this way results known in one space dimension. Our main improvement of existing one-dimensional results consists in giving sharp decay rates, provided that the initial data are regular and the damping term is linear. A crucial intermediate step is the proof of a generalization of Ingham's inequality on nonharmonic Fourier series
Critical Exponents for the Blowup of Solutions with Sign Changes in a Semilinear Parabolic Equation, II

AbstractThe blowup of solutions of the Cauchy problem[formula]is studied. LetΛkbe the set of functions onRwhich change signktimes. It is shown that forpk=1+2/(k+1),k=0, 1, 2, …, any solution withu0∈Λkblows up in finite time if 1<p⩽pk, whereas a global solution withu0∈Λkexists ifp>pk. This is an extension of our previous result [17], in which a fast decay condition was imposed on initial data. It is also shown in this paper that if u0 decays more slowly than |x|−2/(p−1)as |x|→+∞, then the solution blows up in finite time regardless of the number of sign changes
Stability of Fronts for a KPP-System, II: The Critical Case

AbstractWe study a system ofn+1 coupled semilinear parabolic equations on the real line, which depends on a small parameterλand reduces to the scalar Kolmogorov–Petrovsky–Piscounov (KPP) equation, whenλ=0. Under appropriate scaling, the system has a family of traveling fronts, parametrized by their speedγ, when |γ|⩾2, as in the scalar KPP case. The case of critical speed,γ=−2 say, is investigated and it is shown that the system inherits some crucial properties of the KPP equation, whenλis small: in particular, the asymptotic stability of the front in a local and semiglobal sense. First, we describe the properties of the front and then apply functional arguments to prove its local stability in an adequate weighted Sobolev space. Moreover, the decay rate of the perturbations is shown to be polynomial in time. Finally we show a semiglobal stability property of the front, which also is inherited from the scalar KPP equation
A Neumann Problem with Asymmetric Nonlinearity and a Related Minimizing Problem

AbstractVariational methods are used to prove the existence of solution of some classes of nonlinear ordinary differential equations with zero Neumann boundary conditions. Among other very mild conditions, the nonlinearities we consider have a behavior at −∞ at least like a linearity of slope less thanπ2/4. A variational characterization of this valuevis used to prove both geometrical and compactness properties of the associated Euler functional. A partial differential equation version of these problems is also considered
Linearization and Special Asymptotic Behaviour of the Second Painlevé Equation

AbstractLinearization of the initial value problem associated with the special second Painlevé equation is discussed using a different isomonodromic spectral problem than the one used in [1]. Further properties of the monodromy data [2, 3] are detected and these properties are used to reduce the problem to a linear singular integral equation via a Riemann–Hilbert boundary value problem on an imaginary line. A special asymptotic increasing solution of the Painlevé equation is constructed forx→−∞ from the above integral equation. Moreover, failure to extract asymptotics forx→+∞ is also mentioned
Approximation Dynamics and the Stability of Invariant Sets

AbstractWe introduce the concept of a weakly, normally hyperbolic set for a system of ordinary differential equations. This concept includes the notion of a hyperbolic flow, as well as that of a normally hyperbolic invariant manifold. Moreover, it has the property that it is closed under finite set products. Consequently, the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly, normally hyperbolic sets. Our main objective is to show that under a smallC1-perturbation, a weakly, normally hyperbolic setKis preserved by a homeomorphism, where the imageKYis a compact invariant set, with a related hyperbolic structure, for the perturbed equation. In addition, the homeomorphism is close to the identity inC0,1and the perturbed dynamics onKYare close to the original dynamics onK
A Characterization of theω-Limit Sets of Planar Continuous Dynamical Systems

AbstractIf the family of curves G={Γi}iand the set of pointsSare given, we find necessary and sufficient conditions for the existence of aCkmapffrom the plane into itself, 0⩽k⩽∞ (ifk=0 we also assume thatfis locally Lipschitz), such that the curves from G and the points fromSare respectively nondegenerate trajectories and singular points of the systemx′=f(x), and additionallyS∪(∪iΓi) is theω-limit set of some trajectory of the system. In such cases, we also provide a detailed geometric description ofS∪(∪iΓi) (after contraction of the connected components ofS)
Nonresonance Conditions for Asymptotically Positively Homogeneous Differential Systems: The Fučik Spectrum and Its Generalization

AbstractIn this article we propose how to define the Fučik spectra for differential systems in Rnwith jumping nonlinearities. The PropertyP, a generalization of the non-Fučik spectra to positively homogeneous nonautonomous differential systems, is also defined in the most general setting. A quantitative estimate result concerning the parameterized completely continuous, positively homogeneous perturbations of the identity in Banach spaces is given in this article. Such a result is a generalization of one property in the spectrum theory for completely continuous linear operators. Using this estimate result, we will give some significantly improved nonresonance results from the Fučik spectra and PropertyPfor asymptotically positively homogeneous differential systems
α-Hölder Linearization

AbstractA well known theorem of Hartman and Grobman says that aC2diffeomorphismf:Rn→Rnwith a hyperbolic fixed point at 0 can be topologically conjugated to the linear diffeomorphismL=df(0) (in a neighborhood of 0). On the other hand, if a non-planar map has resonance, then linearization may not beC1. A counter-example is due to P. Hartman (see [H2]). In this paper we will show that for anyα∈(0, 1) there exists anα-Hölder linearization in a neighborhood of 0 for the counterexample of Hartman. No resonance condition will be required. A linearization of a more general map will be discussed
Existence of the Entropy Solution for a Viscoelastic Model

AbstractIn this paper, we consider the Cauchy problem for a viscoelastic model with relaxationut+σx=0,(σ−f(u))t+1δ(σ−μf(u))=0with discontinuous, large initial data, where 0<μ<1,δ>0 are constants. When the system is nonstrictly hyperbolic, under the additional assumptionv0x∈L∞, the system is reduced to an inhomogeneous scalar balance law by employing the special form of the system itself. After introducing a definition of entropy solutions to the system, we prove the existence, uniqueness, and continuous dependence of the global entropy solution for the system. When the system is strictly hyperbolic, some special entropy pairs of the Lax type are constructed, in which the progression terms are functions of a single variable, and the necessary estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations. The special entropy pairs are used to prove the existence of the global entropy solutions for the corresponding Cauchy problem by applying the method of compensated compactnes
The Operator EquationAX−XB=C, Admissibility, and Asymptotic Behavior of Differential Equations

AbstractIt is shown that there exists a natural relationship between the regular admissibility of a translation-invariant subspace M ofBUC(R, E) (the space of uniformly continuous bounded functions onRwith values in a Banach spaceE), w.r.t. the differential equationu′(t)=Au(t)+f(t) (∗), and the unique solvability of a special operator equation of Lyapunov's typeAX−XDM=−δ0, where DMis the restriction of the operator D≡d/dtonto M andδ0(f)=f(0). This leads to some spectral conditions for the regular admissibility. Applications to questions of exponential dichotomy, exponential stability, the existence of periodic and almost periodic solutions of equation (∗), as well as analogous questions for some nonlinear equations and functional differential equations are presented
On the Conformal Gaussian Curvature Equation inR2

AbstractIn this paper, we consider the equationΔu+K(x)e2u=0inR2(0.1)whereK(x)=K(|x|) inR2andK(x) does not decay at |x| large. From the geometric viewpoint, it is quite interesting to ask what the best possible range is of total curvature of all solutions of (0, 1). In this paper, we study this problem for radial solutions. We also construct some particularKto demonstrate the rich phenomenon. In particular, we show by examples that whenK(x) is negative for |x| large, Eq. (0.1) possess a branch of solutions which satisfies some monotonicity property
One Symmetry Does Not Imply Integrability

AbstractWe show that Bakirov's counter-example (which had been checked by computer algebra methods up to order 53) to the conjecture that one nontrivial symmetry of an evolution equation implies infinitely many is indeed a counter-example. To prove this we use thesymbolic methodof Gel'fand–Dikii andp-adic analysis. We also formulate a conjecture to the effect that almost all equations in the family considered by Bakirov have at most finitely many symmetries. This conjecture depends on the solution of a diophantine problem, which we explicitly state
Global BV Solutions of Compressible Euler Equations with Spherical Symmetry and Damping

AbstractIn this paper, we study the global existence of BV solutions for compressible Euler equations with spherical symmetry and damping, using Glimm's scheme. The key point to consider is the combination of the geometric effects due to the spherical symmetry and the effects of the frictional damping on the total variation of the solutions. By measuring the strength of the waves byr-s, where (r, s) are the Riemann invariants, we construct a function which leads to the boundedness of the BV norm of the corresponding approximate solutions
Pseudo-spherical Surfaces and Integrability of Evolution Equations

AbstractA complete classification of evolution equationsut=F(x,t,u,ux,…,uxk) which describe pseudo-spherical surfaces, is given, thus providing a systematic procedure to determine a one-parameter family of linear problems for which the given equation is the integrability condition. It is shown that for every second-order equation which admits a formal symmetry of infinite rank (formalintegrability) such a family exists (kinematicintegrability). It is also shown that this result cannot be extended as proven to third-order formally integrable equations. This fact notwithstanding, a special case is proven, and moreover, several equations of interest, including the Harry–Dym, cylindrical KdV, and a family of equations solved by inverse scattering by Calogero and Degasperis, are shown to be kinematically integrable. Conservation laws of equations describing pseudo-spherical surfaces are studied, and several examples are given
On the Steady-State Flow of an Incompressible Fluid through a Randomly Perforated Porous Medium

AbstractAn incompressible fluid is assumed to satisfy the time-independent Stokes equations in a porous medium. The porous medium is modeled by a bounded domain inRnthat is perforated for eachε>0 byε-dilations of a subset ofRnarising from a family of stochastic processes which generalize the homogeneous random fields. The solution of the Stokes equations on these perforated domains is homogenized asε→0 by means of stochastic two-scale convergence in the mean and the homogenized limit is shown to satisfy a two-pressure Stokes system containing both deterministic and stochastic derivatives and a Darcy-type law which generalizes the Darcy law obtained for fluid flow in periodically perforated porous media