pinecone/core-2020-05-10-deduplication huggingface dataset:
Existence and Bifurcation of the Positive Solutions for a Semilinear Equation with Critical Exponent

AbstractIn this paper, we consider the semilinear elliptic equation[formula]Forp=2N/(N−2), we show that there exists a positive constantμ*>0 such that (∗)μpossesses at least one solution ifμ∈(0, μ*) and no solutions ifμ>μ*. Furthermore, (∗)μpossesses a unique solution whenμ=μ*, and at least two solutions whenμ∈(0, μ*) and 2<N<6. ForN⩾6, under some monotonicity conditions onf((1.6)) we show that there exist two constants 0<μ**⩽μ**<μ* such that problem (∗)μpossesses a unique solution forμ∈(0, μ**), and at least two solutions ifμ∈(μ**, μ*)
Regular Solutions of Second-Order Stationary Hamilton–Jacobi Equations

AbstractWe study a second order stationary Hamilton–Jacobi equation in infinite dimension. This equation is nonlinear and convex with respect to the first-order term. We use properties of the transition semigroup associated to the linear equation to write the Hamilton–Jacobi equation in integral form and we prove existence, uniqueness and regularity of a solution by the theory of maximal monotone operators. We also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems. Finally, the solution is the value function of the associated optimal stochastic control problem. Some examples are given
The Existence of Transverse Homoclinic Solutions for Higher Order Equations

AbstractDifferential equations are considered which contain a small parameter. When the parameter is zero the equation is autonomous with a hyperbolic equilibrium and a homoclinic solution. No restriction is placed on the dimension of the phase space or the dimension of intersection of the stable and unstable manifolds. Sufficient conditions on the perturbation terms for producing a transverse homoclinic solution and hence chaos are obtained. The technique follows the work of Palmer using exponential dichotomies and the method of Lyapunov–Schmidt
Semilinear Periodic-Parabolic Equations with Nonlinear Boundary Conditions

AbstractRecently much work has been devoted to periodic-parabolic equations with linear homogeneous boundary conditions. However, very little has been accomplished in the literature for periodic-parabolic problems with nonlinear boundary conditions. It is the purpose of this paper to prove existence and regularity results for (classical) periodic solutions to semilinear second order parabolic partial differential equations with nonlinear boundary conditions provided ordered upper and lower solutions are given. Fractional order function spaces, Ehrling–Gagliardo–Nirenberg and Lions–Peetre–Calderón type interpolation inequalities for functions in (anisotropic) Sobolev–Slobodeckii spaces play an important role in the obtainment ofa prioriboundary and interior estimates. In proving our existence results we make use of topological degree techniques and regularity results for linear parabolic partial differential equations under linear nonhomogeneous boundary conditions. We also indicate how one can obtain minimal and maximal time-periodic solutions to parabolic problems with nonlinear boundary conditions
Estimates on Invariant Tori near an Elliptic Equilibrium Point of a Hamiltonian System

AbstractWe give a precise statement for the KAM theorem in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system. If the frequencies of the elliptic point are nonresonant up to a certain orderK⩾4, and a nondegeneracy condition is fulfilled, we get an estimate for the measure of the complement of the KAM tori in a neighbourhood of given radius. Moreover, if the frequencies satisfy a Diophantine condition, with exponentτ, we show that in a neighbourhood of radiusrthe measure of the complement is exponentially small in (1/r)1/(τ+1). We also give a related result for quasi-Diophantine frequencies, which is more useful for practical purposes. The results are obtained by putting the system in Birkhoff normal form up to an appropiate order, and the key point relies on giving accurate bounds for its terms
Fast/Slow Diffusion and Growing Sandpiles

AbstractWe study a coupled system of ODE (introduced by the first author inSIAM J. Appl. Math.22(1972) 437–458) for the heights of growing, interacting sand cones. We show that these ODE correspond to the evolution inL2generated by the subdifferential of the convex functional which vanishes on functions whose gradient has length less than or equal to one and is infinity otherwise. Additionally we explain how the ODE arise from evolutions governed by thep-Laplacian in the “infinitely fast/infinitely slow” diffusion limit asp→∞
On the Existence of the Compact Global Attractor for Semilinear Reaction Diffusion Systems on RN

AbstractWe show that a class of reaction diffusion systems on RNgenerates an asymptotically compact semiflow on the Banach space of bounded uniformly continuous functions. If such a semiflow is dissipative, then a unique, non-empty, compact minimal attractor is known to exist. We apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on RNthat contain appropriate weight functions. We also state conditions, which guarantee that the attractor has finite Hausdorff-dimension
On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

AbstractWe consider a very general second order nonlinear parabolic boundary value problem. Assuming the existence of an upper solutionϕand a lower solutionψsatisfyingψ⩽ϕ, we show that the problem has extremal periodic solutions in the order intervalK=[ψ, ϕ]. Our proof is based on a general surjectivity result for the sum of two operators of monotone type and on truncation and penalization techniques. In addition we use a result of independent interest which we prove here and which says that the pseudomonotonicity property ofA(t, ·) can be lifted to its Nemitsky operator. Finally when we impose stronger conditions on the data, we show that the extremal solutions can be obtained with a monotone iterative process
Stability Results for the Wave Equation with Indefinite Damping

AbstractThe one-dimensional wave equation with damping of indefinite sign in a bounded interval with Dirichlet boundary conditions is considered. It is proved that solutions decay uniformly exponentially to zero provided the damping potential is in theBV-class, has positive average, is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes. This sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms. The method of proof relies on the developments by S. Cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa
Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems

AbstractA formula is given that counts the number of roots in the positive half plane of the characteristic equation for general real, constant coefficient, linear delay-differential systems. The formula is used to establish necessary and sufficient conditions for asymptotic stability of the zero solution of linear delay-differential systems. The formula is potentially useful in verifying stability hypotheses that arise in bifurcation analysis of autonomous delay-differential systems. Application of the formula to Hopf bifurcation theory for delay-differential systems is discussed, and an example application to an equation with two delays is given
Fast and Slow Waves in the FitzHugh–Nagumo Equation

AbstractIt is known that the FitzHugh–Nagumo equation possesses fast and slow travelling waves. Fast waves are perturbations of singular orbits consisting of two pieces of slow manifolds and connections between them, whereas slow waves are perturbations of homoclinic orbits of the unperturbed system. We unfold a degenerate point where the two types of singular orbits coalesce forming a heteroclinic orbit of the unperturbed system. Letcdenote the wave speed andεthe singular perturbation parameter. We show that there exists aC2smooth curve of homoclinic orbits of the form (c, ε(c)) connecting the fast wave branch to the slow wave branch. Additionally we show that this curve has a unique non-degenerate maximum. Our analysis is based on a Shilnikov coordinates result, extending the Exchange Lemma of Jones and Kopell. We also prove the existence of inclination-flip points for the travelling wave equation thus providing the evidence of the existence ofn-homoclinic orbits (n-pulses for the FitzHugh–Nagumo equation) for arbitraryn
Analyticity of Solutions for a Generalized Euler Equation

AbstractWe consider the so-called lake and great lake equations, which are shallow water equations that describe the long-time motion of an inviscid, incompressible fluid contained in a shallow basin with a slowly spatially varying bottom, a free upper surface, and vertical side walls, under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude. If these equations are posed on a space-periodic domain and the initial data are real analytic, the solution remains real analytic for all times. The proof is based on a characterization of Gevrey classes in terms of decay of Fourier coefficients. In particular, our result recovers known results for the Euler equations in two and three spatial dimensions. We believe the proof is new
Spectral Theory of Laplace–Beltrami Operators with Periodic Metrics

AbstractThe spectra of Laplace–Beltrami operators with periodic metrics has been less investigated than that of Schrödinger operators with a periodic potentials, and there are many differences between these two cases. It has been established that the spectrum of a Laplace–Beltrami operator with periodic metric is the union of closed intervals and that spectral gaps are possible, but whether an infinite number of spectral gaps is possible is an open question. Utilizing special transformations it is shown that there are two-dimensional Laplace–Beltrami operators having an arbitrarily large number of spectral gaps by proving that a particular two-dimensional operator has an infinite number of spectral gaps. In the case of a periodic conformal metric relationships between the scalar curvature and the number of gaps are investigated
A Pointwise Lower Bound for Positive Solutions of a Schrödinger Equation in RN

AbstractStrong positivity of the bounded inverse (−A)−1of a Schrödinger operator −A=−Δ+q(x)• inL2(RN) is proved in the following form: If −Au=f⩾0 inL2(RN) withf≢0, thenu⩾cϕ1a.e. in RN. Here,ϕ1is the positive eigenfunction associated with the principal eigenvalueλ1of −A, andcis a positive constant. It is shown that this result is valid if and only if the potentialq(x), which is assumed to be strictly positive and locally bounded, has a sufficiently fast growth as |x|→∞. This result is applied to linear and nonlinear elliptic boundary value problems in strongly ordered Banach spaces, whose positive cone is generated by the eigenfunctionϕ1. In particular, problems of existence and uniqueness are addressed
Local Minimizers Induced by Spatial Inhomogeneity with Inner Transition Layer

AbstractThe main aim of this paper is to construct and provide the structure of local minimizers of the variational problem infv∈L1(Ω)Fε(v), whereε: small parameter and[formula]having only the functionsk1>0 andk2>0 as parameters. Given any closed simple smooth curveγmcontained in a level set ofa(X)=k1(X)k2(X),X∈Ω, we give necessary conditions relating the slope and concavity ofa(X),X∈γm, along the normaln[γm(X)], with the curvatureκ(X) ofγmso that the above problem possesses a nonconstant minimizervε(X) whose nodal curveγεmconverges uniformly toγm, asε→0. Moreovervε→1, asε→0, uniformly on compact sets contained in one of the two connected components ofΩ\γmandvε→−1, asε→0, uniformly on compact sets contained in the other one
On the Linear Damped Wave Equation

AbstractIn this work we estimate the spectrum of the linear damped wave semigroup under homogeneous Dirichlet boundary conditions by using the principal eigenvalue of an elliptic operator related to the equation. Our estimate is optimal for real eigenvalues. Then, we analyze the behavior of the estimate as thedamping amplitudegrows to infinity. When the damping changes of sign we extend a result of Freitas [5] to show that the semigroup possesses at least two real eigenvalues greater than one if the amplitude is sufficiently large. In particular, the trivial state is unstable, in strong contrast with the sign definited case. Finally, we characterize the limiting behavior of the real eigenpairs which originate the inestability of the trivial state. This analysis is based upon the behavior of the principal eigenpair of a singular perturbation problem at the singular limit. Our theory is of interest by itself and it has many applications to reaction diffusion systems (c.f. for instance [6])
Darboux Linearization and Isochronous Centers with a Rational First Integral

AbstractIn this paper we study isochronous centers of polynomial systems. It is known that a center is isochronous if and only if it is linearizable. We introduce the notion of Darboux linearizability of a center and give an effective criterion for verifying Darboux linearizability. If a center is Darboux linearizable, the method produces a linearizing change of coordinates. Most of the known polynomial isochronous centers are Darboux linearizable. Moreover, using this criterion we find a new two-parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family. We also determine all Hamiltonian cubic systems which are Darboux linearizable. In the second part of this work we restrict to the study of isochronous centers having a rational first integral. We prove that, under certain conditions, the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber, or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center
Global Existence and Asymptotic Behavior in Nonlinear Thermoviscoelasticity

AbstractWe study global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of weak solutions to the system of nonlinear thermoviscoelasticity. Various boundary conditions are considered. It is shown that for any initial data (u0, v0, θ0)∈L∞×W1, ∞×H1there is a unique global solution (u, v, θ)=(deformation gradient, velocity, temperature) such thatu∈C([0, ∞], L∞),v∈C((0, ∞), W1, ∞)∩L∞([0, ∞), W1, ∞),θ∈C([0, ∞), H1). The constitutive assumptions for the Helmholtz free energy include the models for the study of phase transition problems in shape memory alloys
Asymptotic Behavior for Scalar Viscous Conservation Laws with Boundary Effect

AbstractWe consider the asymptotic stability of viscous shock waveφfor scalar viscous conservation lawsut+f(u)x=uxxon the half-space (−∞, 0) with boundary valuesu∣x=−∞=u−,u∣x=0=u+. Our problem is divided into three cases depending on the sign of shock speedsof the shock (u−, u+). Whens⩽0, the asymptotic state ofubecomesφ(·+d(t)), whered(t) depends implicitly on the initial datau(x, 0) and is related to the boundary layer of the solution at the boundaryx=0. The stability of this state fors<0 will be shown by applying the weighted energy method. Fors=0 a conjecture ond(t) will be presented. The cases>0 is also treated
Decay Rate for Travelling Waves of a Relaxation Model

AbstractA relaxation model was proposed in [Shi Jin and Zhouping Xin,Comm. Pure Appl. Math.48(1995), 555–563] to approximate the hyperbolic systems numerically under the subcharacteristic condition introduced in [T. P. Liu,Comm. Math. Phys.108(1987), 153–175]. The stability of travelling waves with strong shock profile and integral zero was proved in [H. L. Liu, J. H. Wang, and T. Yang, Stability in a relaxation model with nonconvex flux, preprint, 1996; H. L. Liu and J. Wang, Asymptotic stability of travelling wave solutions of a hyperbolic system with relaxation terms, preprint, 1995] when the original system is scalar. In this paper, we study the rate of the asymptotic convergence speed of thse travelling wave solutions. The analysis applies to the case of a nonconvex flux and when the shock speed coincides with characteristic speed of the state at infinity. The decay rate is obtained by applying the energy method and is shown to be the same as the one for the viscous conservation law [A. Matsumura and K. Nishihara,Comm. Math. Phys.165(1994), 83–96]
A Geometric Method for Detecting Chaotic Dynamics

AbstractA new method of detection of chaos in dynamical systems generated by time-periodic nonautonomous differential equations is presented. It is based on the existence of some sets (called periodic isolating segments) in the extended phase space, satisfying some topological conditions. By chaos we mean the existence of a compact invariant set such that the Poincaré map is semiconjugated to the shift on two symbols and the counterimage (by the semiconjugacy) of any periodic point in the shift contains a periodic point of the Poincaré map. As an application we prove that the planar equationż=(1+eiφt |z|2) zgenerates chaotic dynamics provided 0<φ⩽1/288
Instability of Discontinuous Traveling Waves for Hyperbolic Balance Laws

AbstractIt is known that scalar hyperbolic conservation laws with source term and periodic initial value have a property of Poincaré–Bendixson type, namely the solutions converge either to a constant state or to a periodic traveling wave, which is necessarily discontinuous. In this paper we show that generically (with respect to theL1topology) the solutions exhibit a behaviour of the former type. We also show that, while the rate of convergence to a constant state is exponential, the convergence to a traveling wave can be arbitrarily slow
Topological Multivortex Solutions of the Self-Dual Maxwell–Chern–Simons–Higgs System

AbstractWe study the existence and various behaviors of topological multivortex solutions of the relativistic self-dual Maxwell–Chern–Simons–Higgs system. We first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional. Then, by an iteration method, we prove the existence of topological solutions satisfying some extra conditions, which we call admissible solutions. We establish asymptotic exponential decay estimates for these topological solutions. We also investigate the limiting behavior of the admissible solutions as parameters in our system go to some limits. For the Abelian Higgs limit we obtain strong convergence result, while for the Chern–Simons limit we only obtained that our admissible solutions weakly approximate one of the Chern–Simons solutions
Stability of Traveling Waves of a Solute Transport Equation

AbstractWe prove the large time asymptotic stability of traveling wave solutions to the scalar solute transport equation (contaminant transport equation) with spatially periodic diffusion-adsorption coefficients in one space dimension. The time dependent solutions converge in proper norms to a translate of traveling wave solutions as time approaches infinity. In case of classical traveling waves, the convergence rate is exponential in time for a class of small initial perturbations; and for general order one perturbations, the convergence holds in supremum norm. In case of degenerate Hölder continuous traveling waves, the convergence holds inL1norm. As a byproduct, uniqueness up to translation of degenerate traveling waves follows. We use maximum principle,L1contraction, spectral theory, and a space-time invariance property of solutions
Homoclinics and Heteroclinics for a Class of Conservative Singular Hamiltonian Systems

AbstractWe consider an autonomous Hamiltonian systemü+∇V(u)=0 where the potentialV:R2\{ξ}→Rhas a strict global maximum at the origin and a singularity at some pointξ≠0. Under some compactness conditions onVat infinity and around the singularityξwe study the existence of homoclinic orbits to 0 winding aroundξ. We use a sufficient, and in some sense necessary, geometrical condition (∗) onVto prove the existence of infinitely many homoclinics, each one being characterized by a distinct winding number aroundξ. Moreover, under the condition (∗) there exists a minimal non contractible periodic orbitūand we establish the existence of a heteroclinic orbit from 0 toū. This connecting orbit is obtained as the limit in theC1loctopology of a sequence of homoclinics with a winding number larger and larger
Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

AbstractWe discuss the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations. Our approach is via monotone iteration and non-standard ordering in the profile set for asymptotic boundary value problems and viaS1-degree and equivariant bifurcation theory for periodic boundary value problems. Applications will be given to wave fronts and to slowly oscillatory spatially periodic traveling waves of lattice delay differential equations arising from population genetics, population dynamics, and neural networks
Asymptotic First Eigenvalue Estimates for the Biharmonic Operator on a Rectangle

AbstractWe find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small
On Spectral Properties of the Acoustic Propagator in a Layered Band

AbstractThe acoustic propagator is the self-adjoint operatorH=−∇·c(x)2∇, defined inL2(Q), whereQ⊆R2is a band of finite width, given byQ={(x, z), x∈R, 0⩽z⩽Γ}. The wave velocitycdepends only on the horizontal coordinatex, and is a measurable bounded function, converging toc±>0 asx→±∞. It is proved that the resolvent operatorR(z)=(H−z)−1, Imz≠0, can be extended continuously to the closed upper (or lower) half-plane, in a suitable weighted-L2topology (“Limiting Absorption Principle”). In particular, this continuity holds at the thresholdλ=0. It follows as a corollary thatHhas no point spectrum
Histogram Modification via Differential Equations

AbstractThe explicit use of partial differential equations (PDEs) in image processing became a major research topic in the past years. In this work we present a framework for histogram (pixel-value distribution) modification via ordinary and partial differential equations. In this way, the image contrast is improved. We show that the histogram can be modified to achieve any given distribution as the steady state solution of an image flow. The contrast modification can be performed while simultaneously reducing noise in a unique PDE, avoiding noise sharpening effects of classical algorithms. The approach is extended to local contrast enhancement as well. A variational interpretation of the flow is presented and theoretical results on the existence of solutions are given