pinecone/core-2020-05-10-deduplication huggingface dataset:
Remarks on Third-Order ODEs Relevant to the Kuramoto–Sivashinsky Equation

AbstractWe are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto–Sivashinsky equation. So-called steady solutions of the Kuramoto–Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited
Spikes for the Gierer–Meinhardt System in Two Dimensions: The Strong Coupling Case

AbstractNumerical computations often show that the Gierer–Meinhardt system has stable solutions which display patterns of multiple interior peaks (often also called spots). These patterns are also frequently observed in natural biological systems. It is assumed that the diffusion rate of the activator is very small and the diffusion rate of the inhibitor is finite (this is the so-called strong-coupling case). In this paper, we rigorously establish the existence and stability of such solutions of the full Gierer–Meinhardt system in two dimensions far from homogeneity. Green's function together with its derivatives plays a major role
Mathematical Analysis of Geophysical Balance Models

AbstractAn analysis of an approximation to the rotating shallow-water equations is presented. The approximation removes the fast waves without introducing secular terms and is valid for physical boundaries and prepared initial data. In particular, the shallow-water equations are decomposed into two equations describing the slow and fast dynamics. The basic idea is one of enslaving in which the fast part of the solution is expressed as a function of the slow part yielding an approximation to the slow dynamics. Existence and convergence theorems are given
Global Solution of an Initial-Value Problem for Two-Dimensional Compressible Euler Equations

AbstractThis paper is concerned with the existence of global continuous solutions of the expansion of a wedge of gas into a vacuum for compressible Euler equations. By hodograph transformation, we first prove that the flow is governed by a partial differential equation of second order, which is further reduced to a system of two nonhomogeneous linearly degenerate equations in the phase space under an irrotationality condition. Then this conclusion is applied to solving the problem that a wedge of gas expands into a vacuum, which is actually a Goursat-type problem for these two equations in the supersonic domain
An Lp Theory of Invariant Manifolds for Parabolic Partial Differential Equations on Rd

AbstractWe study the problem about the existence of finite-dimensional invariant manifolds for nonlinear heat equations of the form∂u/∂τ=▵u+F(u, ∇u)onRd×[1, ∞). We show that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero, there exist finite dimensional invariant manifolds which control the long time asymptotics of solutions. We consider the problem for these equations in the framework of weighted Sobolev spaces of Lp type. The Lp theory of this problem gives the L∞ estimate of the long-time asymptotics of solutions under natural assumptions on the nonlinear term F and their initial data
Mini-Maximizers for Reaction-Diffusion Systems with Skew-Gradient Structure

AbstractA reaction-diffusion system with skew-gradient structure is a sort of activator-inhibitor system that consists of two gradient systems coupled in a skew-symmetric way. Any steady state of such a system corresponds to a critical point of some functional. The aim of this paper is to study the relation between a stability property as a steady state of the reaction-diffusion system and a mini-maximizing property as a critical point of the functional. It is shown that a steady state of the skew-gradient system is stable regardless of time constants if and only if it is a mini-maximizer of the functional. It is also shown that the mini-maximizing property is closely related with the diffusion-induced instability. Moreover, by using the property that any mini-maximizer on a convex domain is spatially homogeneous, quite a general instability criterion is obtained for some activator-inhibitor systems. These results are applied to the diffusive FitzHugh–Nagumo system and the Gierer–Meinhardt system
A Local Bifurcation Theorem for Degenerate Elliptic Equations With Radial Symmetry

AbstractIn this work we provide local bifurcation results for equations involving the p-Laplacian in balls. We analyze the continua Cn of radial solutions emanating from (λn, p, 0), {λn, p} being the radial eigenvalues of −Δp. First, we show that the only nontrivial solutions close to (λn, p, 0) lie on a continuous curve, thus extending the Crandall–Rabinowitz theorem. Second, it is proved that Cn\{(λn, p, 0)} splits into two unbounded connected pieces, characterized by their nodal properties thus sharpening previous results
Asymptotic Control and Stabilization of Nonlinear Oscillators with Non-isolated Equilibria

AbstractLet Φ:H→R be a C1 function on a real Hilbert space H and let γ>0 be a positive (damping) parameter. For any control function ε:R+→R+ which tends to zero as t→+∞, we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator(HBFC)x(t)+γx(t)+∇Φ(x(t))+ε(t)x(t)=0. We show that if ε(t) does not tend to zero too rapidly as t→+∞, then the term ε(t)x(t) asymptotically acts as a Tikhonov regularization, which forces the trajectories to converge to a particular equilibrium. Indeed, in the main result of this paper, it is established that, when Φ is convex and S=argminΦ≠∅, under the key assumption that ε is a “slow” control, i.e., ∫+∞0ε(t)dt=+∞, then each trajectory of the (HBFC) system strongly converges, as t→+∞, to the element of minimal norm of the closed convex set S. As an application, we consider the damped wave equation with Neumann boundary condition[formula
Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

AbstractWe are concerned with singular limits of stiff relaxation and dominant diffusion for general 2×2 nonlinear systems of conservation laws, that is, the relaxation time τ tends to zero faster than the diffusion parameter ε, τ=o(ε), ε→0. We establish the following general framework: If there exists an a priori L∞ bound that is uniformly with respect to ε for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other type of limits. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L∞ bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in Lp to the equilibrium solutions of these systems is established, provided that the relaxation time τ tends to zero faster than ε
Area-Preserving Normalizations for Centers of Planar Hamiltonian Systems

AbstractIt is well known that a nondegenerate center of an analytic Hamiltonian planar system can be brought to normal form by means of an analytic canonical change of coordinates. This normal form, that we denote by CNF, does not depend on the coordinate transformation. In this paper we give an elementary proof of these facts and we show some interesting applications of the machinery that we develop in order to prove them. For instance, we describe the space of coordinate transformations that bring a Hamiltonian nondegenerate center to its CNF, and we prove that they are all canonical when the center is non-isochronous. We also show that two Hamiltonian systems with a nondegenerate center are canonically conjugated if and only if both centers have the same period function
Boundary Layers Associated with Incompressible Navier–Stokes Equations: The Noncharacteristic Boundary Case

AbstractThe goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier–Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209–230) where the convergence in L2 of the solutions of the Navier–Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier–Stokes equations for incompressible fluids
Hamiltonian Systems near Relative Equilibria

AbstractWe give explicit differential equations for the dynamics of Hamiltonian systems near relative equilibria. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems
On Bounded Solutions of the Emden–Fowler Equation in a Semi-cylinder

AbstractBounded solutions of the Emden–Fowler equation in a semi-cylinder are considered. For small solutions the asymptotic representations at infinity are derived. It is shown that there are large solutions whose behavior at infinity is different. These solutions are constructed when some inequalities between the dimension of the cylinder and the homogeneity of the nonlinear term are fulfilled. If these inequalities are not satisfied then it is proved, for the Dirichlet problem, that all bounded solutions tend to zero and have the same asymptotics as small solutions
Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation: An Improvement

AbstractIn this paper, we consider a discrete delay problem with negative feedback x(t)=f(x(t), x(t−1)) along with a certain family of time discretizations with stepsize 1/n. In the original problem, the attractor admits a nice Morse decomposition. We proved in (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36–78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995, J. Dynam. Differential Equations7, 141–190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36–78) we proved certain continuity results about the individual Morse sets, including that if f(x, y)=f(y), then the individual Morse sets are upper semicontinuous at n=∞. In this paper we extend this result to the general case; that is, we prove for general f(x, y) with negative feedback that the Morse sets are upper semicontinuous
Oscillation Theory of Linear Systems

AbstractA function f:R+→Rstrongly oscillates if for every t∈R+ such that f(t)≠0 there exists s>t with f(t)f(s)<0. Let T be a strongly continuous semigroup with generator A in a real Banach space X. We say that a point x∈X strongly oscillates if for every ξ∈X* the function t→ξ(T(t)x) strongly oscillates. If the spectrum of the generator A of semigroup T has no nonnegative real values then almost all points in X strongly oscillate. In the case when A generates a strongly continuous group under the above assumption all points strongly oscillate. We apply the above results to strong oscillation of solutions of difference, differential and functional differential equations
On the Structure of Nilpotent Normal Form Modules

AbstractThe set of vector fields that are in normal form with respect to a given linear part has the structure of a module and is best described by giving the Stanley decomposition of that module. An algorithm is presented that produces a Stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field, given a Stanley decomposition for the ring of invariants. This reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm. Both the inner product normal form (Elphick) and the sl(2) normal form (Cushman–Sanders) are covered within a single theory, and simplified (non-equivariant) versions of each normal form are presented
Cauchy Problem for Evolution Equations of Schrödinger Type

AbstractIn (R. Agliardi, 1995, Internat. J. Math.6, 791–804) we proved the well-posedness of the Cauchy problem in H∞ for some p-evolution equations (p⩾1) with real characteristic roots. For this purpose some assumptions on the lower order terms are needed, which, in the special case p=1, recapture well-known results for hyperbolic operators. In (R. Agliardi, 1995, Internat. J. Math.6, 791–804) the leading coefficients are assumed to be constant. In this paper we allow them to be variable. Our result is applicable to 2-evolution differential operators with real characteristics, i.e., to Schrödinger type operators. This class of operators comprehends, for example, Schrödinger operator Dt−Δx or the plate operator D2t−Δ2x. The Cauchy problem in H∞ for such evolution operators has been studied extensively by Takeuchi when the coefficients in the principal part are constant and the characteristic roots are distinct
Asymptotic Behavior of Solutions for p-System with Relaxation

AbstractIn this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxationvt−ux=0,ut+p(v)x=1ε(f(v)−u), (E) with initial data(v, u)(x, 0)=(v 0(x), u 0(x))→(v±, u±), v±>0,asx→± ∞. (I) We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v±, u±) of the initial data at x=±∞ do not satisfy the equilibrium equation; i.e., u±≠f(v±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2×2 hyperbolic conservation laws with relaxation
Admissible Riemann Solvers for Genuinely Nonlinear p-Systems of Mixed Type

AbstractWe consider in this paper the Riemann problem for p-systems of mixed type that define two hyperbolic phases with a stress function satisfying general genuinely nonlinear hypotheses. We describe here all the global Riemann solvers that are continuous for the L1 distance with respect to initial data while conserving the natural symmetry properties of the p-system and coinciding with the Lax solution when defined: these Riemann solvers can be described entirely by a kinetic function, used to select a manifold of subsonic phase transitions and a corresponding set of supersonic phase transitions
Normal Forms for Perturbed Keplerian Systems

AbstractPerturbations of rotating and nonrotating Keplerian systems with two and three degrees of freedom are considered. The dominant (unperturbed) part of the Hamilton function is the sum of the two-body part and the Coriolis term. The flows associated to these two components are used to put the whole system into normal form by means of symplectic transformations. Three different situations are analysed: (i) the two-body effect dominates the Coriolis part (slow rotations), (ii) the Coriolis effect is stronger than the two-body Hamiltonian (very fast rotations), and (iii) the two effects are comparable (moderate rotations). After performing the transformations to normal form, the system is reduced by one or two degrees of freedom. We describe the reduced phase spaces by calculating the invariants of the symmetry groups associated to the different normal forms (reduced systems). The technique is applied to the reduction of a Hamiltonian system modelling the trapping mechanisms for the electron of a strongly ionized hydrogen atom
Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

AbstractIn this paper we shall consider the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r>0:dX(t)=[−AX(t)+f(t, Xt)]dt+g(t, Xt)dW(t), where we assume that −A is a closed, densely defined linear operator and the generator of a certain analytic semigroup. f:(−∞, +∞)×Cα→H, g:(−∞, +∞)×Cα→L02(K, H) are two locally Lipschitz continuous functions, where Cα=C([−r, 0], D(Aα)), L02(K, H) are two proper infinite dimensional spaces, 0<α<1. Here, W(t) is a given K-valued Wiener process and both H and K are separable Hilbert spaces
Effects of a Degeneracy in the Competition Model Part I. Classical and Generalized Steady-State Solutions

AbstractWe study the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. We show that there exists a critical number λ* such that if λ <λ*, then the model behaves similarly to the well-studied classical competition model where all the coefficient functions are positive constants, but when λ>λ*, new phenomena occur. Our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well. In part I here, we mainly study two kinds of steady-state solutions which determine the dynamics of the model: one consists of finite functions while the other consists of generalized functions which satisfy (u, v)=(∞, 0) on the part of the domain that b(x) vanishes, but are positive and finite on the rest of the domain, and are determined by certain boundary blow-up systems. The research is continued in part II, where these two kinds of steady-state solutions will be used to determine the dynamics of the model
Effects of a Degeneracy in the Competition Model Part II. Perturbation and Dynamical Behaviour

AbstractThis is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. In part I, we mainly discussed the existence of two kinds of steady-state solutions of this system, namely, the classical steady-states and the generalized steady-states. Here we use these solutions to determine the dynamics of the model. We do this with the help of the perturbed model where b(x) is replaced by b(x)+ε, which itself is a classical competition model. This approach also reveals the interesting relationship between the steady-state solutions (both classical and generalized) of the above system and that of the perturbed system
Semigroup Properties and the Crandall Liggett Approximation for a Class of Differential Equations with State-Dependent Delays

AbstractWe present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall–Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations
A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

AbstractWe show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H=(V,(·,·)) that is bounded below generates a continuum of Hilbert spaces {Hr}r>0 and a continuum of self-adjoint operators {Ar}r>0. For reasons originating in the theory of differential operators, we call each Hr the rth left-definite space and each Ar the rth left-definite operator associated with (H,A). Each space Hr can be seen as the closure of the domain D(Ar) of the self-adjoint operator Ar in the topology generated from the inner product (Arx,y) (x,y∈D(Ar)). Furthermore, each Ar is a unique self-adjoint restriction of A in Hr. We show that the spectrum of each Ar agrees with the spectrum of A and the domain of each Ar is characterized in terms of another left-definite space. The Hilbert space spectral theorem plays a fundamental role in these constructions. We apply these results to two examples, including the classical Laguerre differential expression ℓ[·] in which we explicitly find the left-definite spaces and left-definite operators associated with A, the self-adjoint operator generated by ℓ[·] in L2((0,∞);tαe−t) having the Laguerre polynomials as eigenfunctions
Singular Dirac Systems and Sturm–Liouville Problems Nonlinear in the Spectral Parameter

AbstractFor the Dirac operator with spherically symmetric potential V: (0,∞)→R we investigate the problem of whether the boundary points of the essential spectrum are accumulation points of discrete eigenvalues or not. Our main result shows that the accumulation of such eigenvalues is essentially determined by the asymptotic behaviour of V at 0 and ∞. We obtain this result by using a Levinson-type theorem for asymptotically diagonal systems depending on some parameter, a comparison theorem for the principal solutions of singular Dirac systems, and some criteria on the eigenvalue accumulation (respectively, non-accumulation) of λ-nonlinear singular Sturm–Liouville problems
Local Algebras of Differential Operators

AbstractThere is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111–133]. We also make comments on several problems raised by others