pinecone/core-2020-05-10-deduplication huggingface dataset:
Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with a General Boundary Condition

AbstractIn this article we investigate the spectrum and the spectral singularities of the Quadratic Pencil of Schrödinger OperatorLgenerated inL2(R+) by the differential expressionℓ(y)=−y″+[q(x)+2λp(x)−λ2]y,x∈R+=[0,∞)and the boundary condition∫∞0K(x)y(x)dx+αy′(0)−βy(0)=0,wherep,q, and K are complex valued functions, p is continuously differentiable onR+,K∈L2(R+), andα,β∈C, with |α|+|β|≠0. Discussing the spectrum, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditionslimx→∞p(x)=0,supx∈R+{eεx[|q(x)|+|p′(x)|+|K(x)|]}<∞,ε>0.Later we investigate the properties of the principal functions corresponding to the spectral singularities. Moreover, some results about the spectrum ofLare applied to non-selfadjoint Sturm–Liouville and Klein–Gordons-wave operators
Large-Time Behavior of Entropy Solutions of Conservation Laws

AbstractWe are concerned with the large-time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws. We present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions. These approaches allow the solutions of arbitrarily large oscillation without a priori assumption on the ways from which the solutions come. The relation between the large-time behavior of entropy solutions and the uniqueness of Riemann solutions leads to an extensive study of the uniqueness problem. We use a direct method to show the large-time behavior of largeL∞solutions for a class ofm×msystems including a model in multicomponent chromatography; we employ the uniqueness of Riemann solutions and the convergence of self-similar scaling sequence of solutions to show the asymptotic behavior of largeBVsolutions for the 3×3 system of Euler equations in thermoelasticity. These results indicate that the Riemann solution is the unique attractor of large discontinuous entropy solutions, whose initial data areL∞∩L1orBV∩L1perturbation of the Riemann data, for these systems. These approaches also work for proving the large-time behavior of approximate solutions to hyperbolic conservation laws
Realization of Meander Permutations by Boundary Value Problems

AbstractWe consider Neumann boundary value problems of the form uxx+f(x, u, ux)=0 on the unit interval 0⩽x⩽1 for a certain class of dissipative nonlinearities f. Associated to these problems we have (i) meanders in the phase space (u, ux)∈R2, which are connected oriented simple curves on the plane intersecting a fixed oriented line (the u-axis) in n points corresponding to the solutions; and (ii) meander permutations πf∈S(n) obtained by ordering the intersection points first along the u-axis and then along the meander. The meander permutation πf is the permutation defined by the braid of solutions in the space (x, u, ux). It was recently shown by Fiedler and Rocha that πf determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation ut=uxx+f(x, u, ux), up to C0 orbit equivalence. Therefore, these permutations are of considerable importance in the classification problem of the (Morse–Smale) attractors for these dynamical systems. In this paper we present a purely combinatorial characterization of the set of meander permutations that are realizable by the above boundary value problems
Stability Loss Delay in Harvesting Competing Populations

AbstractWhen populations are in competition, it often happens that one of them disappears. Harvesting may be used for the control and management of competing species to stabilize the populations at a persistent equilibrium. A three-dimensional model, where the harvesting effort is a dynamic variable, is studied in the case where the growth rate of the harvesting effort is very slow. The analysis shows that the system can have relaxation oscillations. Dynamic bifurcation theory is used to determine the maximal and minimal values of harvesting effort along this cycle of oscillations
Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source

AbstractIn this paper, we introduce a new method for investigating the rate and profile of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that the rate of blow-up is uniform in all compact subsets of the domain. This results in a flat blow-up profile, except for a boundary layer, whose thickness vanishes astapproaches the blow-up timeT*. In each case, the blow-up rate of |u(t)|∞is precisely determined. Furthermore, in many cases, we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer. The size of the boundary layer then decays likeT*−t, and the solutionu(t,x) behaves like |u(t)|∞d(x)/T*−tin the boundary layer, wheredis the distance to the boundary. Some Fujita-type critical exponents results are also given for the Cauchy problem
On the Structure of the Two-Soliton Interaction for the Korteweg–de Vries Equation

AbstractIn this paper we consider an interaction of two solitons represented by a two-soliton solution of KdV. Unlike previous work on the subject we do not associate solitons with the maxima of the two-soliton solution but with the poles of a certain singular solution which can be considered as a two-antisoliton solution since it annihilates the original two-soliton solution. Motion of these poles may be chosen to represent motion of the solitons participating in the interaction and unlike interaction of the two maxima, interaction of the poles always proceeds in the same manner: The faster one approaches the slower one at a minimal distance, where they exchange identities and then separat
Regular Solutions of the Shabat Equation

AbstractThe Shabat equationf′(t)+q2f′(qt)+f2(t)−q2f2(qt)=μ,t∈R,whereμandqare parameters, is the simplest self-similar reduction of the so-called dressing chain for constructing and analyzing exactly solvable Schrödinger equations. It is so relevant to the study of theq-oscillator algebra in quantum mechanics. The main objective of this paper is to investigate whether the Shabat equation has a nontrivial global solutionf(t)∈C1(R) that is normalizable in the sense that the corresponding potentialu(t)≔f2(t)−f′(t)−μ/(1−q2) is in the function space L1(R) and whether in the caseμ/(1−q2)>0 it has a global solutionf(t)∈C1(R) that is regular in the sense thatf(t)=±μ/(1−q2)+O(t−2) andf′(t)=O(t−3) ast→±∞
Semi-Fredholm Operators and Periodic Solutions for Linear Functional Differential Equations

AbstractWe deal with the inhomogeneous linear periodic equation with infinite delay of the formdx/dt=Ax(t)+B(t,xt)+F(t), whereAis the generator of aC0-semigroup on a Banach space. Assuming that it has a bounded solution, we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space B and in the concrete phase space B=UCg. The key of our approach is the employment of the perturbation theory of semi-Fredholm operators to show that the period map satisfies the condition of the fixed point theorem by Chow and Hale (Funkcial. Ekvac.17(1974), 31–38)
Higher Order Bifurcations of Limit Cycles

AbstractThis paper shows that asymmetrically perturbed, symmetric Hamiltonian systems of the formx=y,y=±(x±x3)+λ1y+λ2x2+λ3xy+λ4x2y,with analyticλj(ε)=O(ε), have at most two limit cycles that bifurcate for smallε≠0 from any period annulus of the unperturbed system. This fact agrees with previous results of Petrov, Dangelmayr and Guckenheimer, and Chicone and Iliev, but shows that the result of three limit cycles for the asymmetrically perturbed, exterior Duffing oscillator, recently obtained by Jebrane and Żoładek, is incorrect. The proofs follow by deriving an explicit formula for thekth-order Melnikov function,Mk(h), and using a Picard–Fuchs analysis to show that, in each case,Mk(h) has at most two zeros. Moreover, the method developed in this paper for determining the higher-order Melnikov functions also applies to more general perturbations of these systems
On the Fundamental Solutions to Stokes Equations

AbstractWe study the behavior of solutions to the stationary Stokes equations near singular points. Employing the power series expansions of harmonic and biharmonic functions, we have local power series expansions of solutions near singular points. Then we find the precise structures of homogeneous solutions near singular points which appear in local power series expansions. From the structures of the homogeneous solutions we characterize the fundamental solutions. Moreover, we study the asymptotic behavior of solutions to Stokes and Navier–Stokes equations under an assumption on directions of velocities
Bridges between the Generalized Sitnikov Family and the Lyapunov Family of Periodic Orbits

AbstractThe linearization of the spatial restricted three–body problem at the collinear equilibrium point L2has two pairs of pure imaginary eigenvalues and one pair of real eigenvalues so the center manifold is four dimensional. By the classical Lyapunov center theorem there are two families of periodic solutions emanating from this equilibrium point. Using normal form techniques we investigate the existence of bridges of periodic solutions connecting these two Lyapunov families. A bridge is a third family of periodic solutions which bifurcates from both the Lyapunov families. We show that for the mass ratio parameterμnear 1/2 and near 0 there are many bridges of periodic solutions
Multiple Internal Layer Solutions Generated by Spatially Oscillatory Perturbations

AbstractFor a singularly perturbed system of reaction-diffusion equations, we study the bifurcation of internal layer solutions due to the addition of a spatially oscillatory term. In the singular limit, the existence and stability of internal layer solutions are determined by the intersection of afast jump surfaceΓ1and aslow switching curveC. The case when the intersection is transverse was studied by X.-B. Lin (Construction and asymptotic stability of structurally stable internal layer solutions, preprint). In this paper, we show that whenΓ1intersects with C tangentially, saddle-node or cusp type bifurcation may occur. Higher order expansions of internal layer solutions and eigenvalue–eigenfunctions are also presented. To find a true internal layer solution and true eigenvalue-eigenfunctions, we use a Newton's method in functions spaces that is suitable for numerical computations
Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications

AbstractWe prove comparison results between viscosity sub- and supersolutions of degenerate elliptic and parabolic equations associated to, possibly nonlinear, Neumann boundary conditions. These results are obtained under more general assumptions on the equation (in particular the dependence in the gradient of the solution) and they allow applications to quasilinear, possibly singular, elliptic or parabolic equations. One of the main applications is the extension of the so-called level set approach for equations set in bounded domains with nonlinear Neumann boundary conditions. In such a framework, the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary
On the Strong Maximum Principle and the Large Time Behavior of Generalized Mean Curvature Flow with the Neumann Boundary Condition

AbstractA generalized mean curvature flow is considered in a cylindrical domain under the right angle boundary condition. Its large time behavior is studied by analyzing the level set equation of the flow when the initial hypersurface is not necessarily graph-like. For a convex domain it is shown that the solution of the level set equation converges to a function whose level sets are perpendicular to the lateral boundary of the domain as time tends to infinity (when the initial data is constant outside some bounded set). For this purpose a version of the strong maximum principle is established for the level set minimal surface equation although the equation is degenerat
Genericity Conditions for Finite Cyclicity of Elementary Graphics

AbstractIn this paper we explore the Khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice. The genericity conditions needed in that case form a proper subset of the usual methods. Moreover some of the conditions are non-intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions. Hence we introduce an extension of the method which treats the usual functional-Pfaffian systems together with the admissible changes of coordinates in the functional equations
Stable Vortex Solutions to the Ginzburg–Landau Equation with a Variable Coefficient in a Disk

AbstractThis paper deals with stable solutions with a single vortex to the Ginzburg–Landau equation having a variable coefficient subject to the Neumann boundary condition in a planar disk. The equation has a positive parameter, say λ, which will play an important role for the stability of the solution. We consider the equation with a radially symmetric coefficient in the disk and suppose that the coefficient is monotone increasing in a radial direction. Then the equation possesses a pair of solutions with a single vortex for large λ. Although these solutions for the constant coefficient are unstable, they can be stable for a suitable variable coefficient and large λ. The purpose of this article is to give a sufficient condition for the coefficient to allow those solutions being stable for any sufficiently large λ. As an application we show an example of the coefficient enjoying the condition, which has an arbitrarily small total variation
Multiple Bounded Solutions of Differential Inclusions: The Nielsen Theory Approach

AbstractThe generalized Nielsen number is defined for self–maps, which are composed by operators with Rδ-values, on compact connected ANRs. Then it is applied to Carathéodory differential inclusions with constraints for obtaining the multiplicity criteria. More precisely, such problems are transformed to those for the lower estimate of fixed points of the related operators with the given properties on bounded, compact, connected neighbourhood retracts of Fréchet spaces. In this way, multiple solutions can be proved e.g. for the multivalued initial value problem on the half–line or, in the single–valued case, for boundary value problems to ordinary differential equations
Singular Perturbations of First-Order Hyperbolic Systems with Stiff Source Terms

AbstractThis work develops a singular perturbation theory for initial-value problems of nonlinear first-order hyperbolic systems with stiff source terms in several space variables. It is observed that under reasonable assumptions, many equations of classical physics of that type admit a structural stability condition. This condition is equivalent to the well-known subcharacteristic condition for one-dimensional 2×2-systems and the well-known time-like condition for one-dimensional scalar second-order hyperbolic equations with a small positive parameter multiplying the highest derivatives. Under this stability condition, we construct formal asymptotic approximations of the initial-layer solution to the nonlinear problem. Furthermore, assuming some regularity of the solutions to the limiting inner problem and the reduced problem, we prove the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval. The stability condition seems to be a key to problems of this kind and can be easily verified. Moreover, this presentation unifies and improves earlier works for some specific equations
Codimension-One Riemann Solutions: Classical Missing Rarefaction Cases

AbstractThis paper is the third in a series that undertakes a systematic investigation of Riemann solutions of systems of two conservation laws in one spatial dimension. Sixty-three codimension-one degeneracies of such solutions have been identified at which strict hyperbolicity is maintained. In this paper, 18 of the degeneracies (9 pairs), constituting the most classical degeneracies, are studied in detail. Precise conditions for a codimension-one degeneracy are identified in each case, as are conditions for folding of the Riemann solution surface, which can occur in 4 of the cases. Such folding gives rise to local multiplicity or nonexistence of Riemann solutions
Normalized Eigenvectors for Nonlinear Abstract and Elliptic Operators

AbstractWe consider the eigenvalue problemAu−C(λ, u)=0for nonlinear operators A, C in a Banach space. The operator A is of type (S)+, or m-accretive, or maximal monotone, while the operator C is compact, or just continuous and bounded, whenever the resolvents of A are compact. The eigenvalues are not necessarily of multiplicative nature (as in the problem Au−λCu=0). Methods are introduced with applications to nonlinear eigenvalue problems involving the existence of eigenvectors with new normalized conditions. Most of the results are new even in the special case of Au−λCu=0
Ergodicity of Dissipative Differential Equations Subject to Random Impulses

AbstractDifferential equations subject to random impulses are studied. Randomness is introduced both through the time between impulses, which is distributed exponentially, and through the sign of the impulses, which are fixed in amplitude and orientation. Such models are particular instances of piecewise deterministic Markov processes and they arise naturally in the study of a number of physical phenomena, particularly impacting systems. The underlying deterministic semigroup is assumed to be dissipative and a general theorem which establishes the existence of invariant measures for the randomly forced problem is proved. Further structure is then added to the deterministic semigroup, which enables the proof of ergodic theorems. Characteristic functions are used for the case when the deterministic component forms a damped linear problem and irreducibility measures are employed for the study of a randomly forced damped double-well nonlinear oscillator with a gradient structure
Boundary Effect on Asymptotic Behaviour of Solutions to the p-System with Linear Damping

AbstractWe consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞),vt−ux=0,ut+p(v)x=−αu, with the Dirichlet boundary condition u|x=0=0 or the Neumann boundary condition ux|x=0=0. The initial date (v0, u0)(x) has the constant state (v+, u+) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599–605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171–188; 137 (1997), 384–395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v+, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v0(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v0(0) and v+. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v0(0)=v+, the optimal convergence rate is also obtained. However, this is not known in the case v0(0)≠v+
Oleinik Type Estimates and Uniqueness for n×n Conservation Laws

AbstractLet ut+f(u)x=0 be a strictly hyperbolic n×n system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of u in a forward neighborhood of each point in the t-x plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleinik in the scalar case
An Extension of Maximum and Anti-Maximum Principles to a Schrödinger Equation in R2

AbstractStrong maximum and anti-maximum principles are extended to weak L2 (R2)-solutions u of the Schrödinger equation −Δu+q(x)u−λu=f(x) in L2(R2) in the following form: Let ϕ1 denote the positive eigenfunction associated with the principal eigenvalue λ1 of the Schrödinger operator A=−Δ+q(x)• in L2(R2). Assume that q(x)≡q(|x|), f is a “sufficiently smooth” perturbation of a radially symmetric function, f≢0 and 0⩽f/ϕ1⩽C≡const a.e. in R2. Then there exists a positive number δ (depending upon f) such that, for every λ∈(−∞, λ1+δ) with λ≠λ1, the inequality (λ1−λ)u⩾cϕ1 holds a.e. in R2, where c is a positive constant depending upon f and λ. It is shown that such an inequality is valid if and only if the potential q(x), which is assumed to be strictly positive and locally bounded, has a superquadratic 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 eigenfunction ϕ1. In particular, problems of existence and uniqueness are addressed
Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems

AbstractWe consider a mild solution u of a well-posed, inhomogeneous, Cauchy problem, u(t)=A(t)u(t)+f(t), on a Banach space X, where A(·) is periodic. For a problem on R+, we show that u is asymptotically almost periodic if f is asymptotically almost periodic, u is bounded, uniformly continuous and totally ergodic, and the spectrum of the monodromy operator V contains only countably many points of the unit circle. For a problem on R, we show that a bounded, uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied. We also show that there is a unique bounded solution subject to certain spectral assumptions on V, f and u
Zero Relaxation Limit to Centered Rarefaction Waves for a Rate-Type Viscoelastic System

AbstractWe study a rate-type viscoelastic system proposed by I. Suliciu (1990, Internat. J. Engrg. Sci.28, 827–841), which is a 3×3 hyperbolic system with relaxation. As the relaxation time tends to zero, this system convergences to the well-known p-system formally. In the case where the initial data are the Riemann data such that the corresponding solutions of the p-system are centered rarefaction waves, we show that if the wave strength is suitably small, then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero, except for an initial layer. The jump discontinuities in the solutions are decaying exponentially fast as time tends to infinity