pinecone/core-2020-05-10-deduplication huggingface dataset:
The Quantizing Effect of Potentials on the Critical Number of Reaction–Diffusion Equations

AbstractWe give an almost complete characterization of the relation between the critical Fujita number of the equation Δu−Vu+up−ut=0 and the potentials V behaving like ±a/(1+d(x)b). Here a>0 and b is any real number. The only type of V not covered is when V(x)∼−a/(1+d(x)b) with b>2 and a is not small. It is interesting to note that when V=±a/(1+d(x)2) we are at a border line case where the critical exponent (depending on a) can vary from 1 to ∞. For all other V (except the ones the theorem does not cover), the critical Fujita number takes only three discrete values: 1, ∞, and 1+2n in the Euclidean case. We also obtain global estimates of Schrödinger heat kernels and a Liouville theorem for a semilinear elliptic equation
Non-resonance Conditions for Semilinear Sturm–Liouville Problems with Jumping Non-linearities

AbstractWe consider the Sturm–Liouville boundary value problem−(p(x)u′(x))′+q(x)u(x)=f(x, u(x))+h(x),  x∈(0, π),c00u(0)+c01u′(0)=0,  c10u(π)+c11u′(π)=0,where p∈C1([0, π]), q∈C0([0, π]), with p(x)>0, x∈[0, π], c2i0+c2i1>0, i=0, 1, h∈L2(0, π), and f:[0, π]×R→R is a Carathéodory function. We assume that the rate of growth of f(x, ξ) is at most linear as |ξ|→∞, but the asymptotic behaviour may be different as ξ→±∞, so the non-linearity is termed “jumping.” Conditions for existence of solutions of this problem are usually expressed in terms of “non-resonance” with respect to the standard Fučı́k spectrum. In this paper we give conditions for both existence and non-existence of solutions in terms of a slightly different idea of the spectrum. These conditions extend the usual Fučı́k spectrum conditions
Asymptotic Behavior of Zeros of Solutions for Parabolic Equations

AbstractLet p>1, α⩾0, and ϕ∈L∞(R)∩L1(R) change its sign finite times. This paper is concerned with a Cauchy problem [formula] Define the set of zeros of a solution u by Z(t)={x∈R:u(x, t)=0} for t>0. In the case of α=0, we show that the set Z(t) is contained in [−Ct, Ct] for large t>0 with some C>0 and that this order of t is best possible. When α>0, we also give estimates of Z(t) for global solutions and prove that Z(t)⊂[−K, K] for all t∈(0, T) with some K>0 for each blowup solution, where T is the blowup time
Boundary Eigenvalue Problems for Differential Equations Nη=λPη with λ-Polynomial Boundary Conditions

AbstractThe present paper deals with the spectral properties of boundary eigenvalue problems for differential equations of the form Nη=λPη on a compact interval with boundary conditions which depend on the spectral parameter polynomially. Here N as well as P are regular differential operators of order n and p, respectively, with n>p⩾0. The main results concern the completeness, minimality, and Riesz basis properties of the corresponding eigenfunctions and associated functions. They are obtained after a suitable linearization of the problem and by means of a detailed asymptotic analysis of the Green's function. The function spaces where the above properties hold are described by λ-independent boundary conditions. An application to a problem from elasticity theory is given
Stability of Singular Equilibria in Quasilinear Implicit Differential Equations

AbstractThis paper addresses stability properties of singular equilibria arising in quasilinear implicit ODEs. Under certain assumptions, local dynamics near a singular point may be described through a continuous or directionally continuous vector field. This fact motivates a classification of geometric singularities into weak and strong ones. Stability in the weak case is analyzed through certain linear matrix equations, a singular version of the Lyapunov equation being especially relevant in the study. Weak stable singularities include singular zeros having a spherical domain of attraction which contains other singular points. Regarding strong equilibria, stability is proved via a Lyapunov–Schmidt approach under additional hypotheses. The results are shown to be relevant in singular root-finding problems
The Motion of a Charged Particle on a Riemannian Surface under a Non-Zero Magnetic Field

AbstractIn this paper we study the motion of a charged particle on a Riemmanian surface under the influence of a positive magnetic field B. Using Moser's Twist Theorem and ideas from classical pertubation theory we find sufficient conditions to perpetually trap the motion of a particle with a sufficient large charge in a neighborhood of a level set of the magnetic field. The conditions on the level set of the magnetic field that guarantee the trapping are local and hold near all non-degenerate critical local minima or maxima of B. Using symplectic reduction we apply the results of our work to certain S1-invariant magnetic fields on R3
Minimizing Nonconvex, Simple Integrals of Product Type

AbstractWe consider the problem of minimizing simple integrals of product type, i.e. min∫T0g(x(t))f(x′(t))dt:x∈AC([0, T]), x(0)=x0, x(T)=xT, (P) where f:R→[0, ∞] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem (P**) obtained from (P) by replacing f with its convex envelope f** admits a solution, we prove attainment for (P) for every continuous, positively bounded below the coefficient g such that (i) every point t∈R is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that, for those f such that the relaxed problem (P**) has a solution, the class of coefficients g that yield existence to (P) is dense in the space of continuous, positive functions on R. We discuss various instances of growth conditions on f that yield solutions to (P**) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp
A Spectral Theory for a λ-Rational Sturm–Liouville Problem

AbstractWe consider the regular Sturm–Liouville problem y″−py+(λ+q/(u−λ))y=0, which contains the eigenvalue parameter rationally. Under certain assumptions on p, q, and u it is shown that the spectrum of the problem consists of a continuous component (the range of the function u), discrete eigenvalues, and possibly a finite number of embedded eigenvalues. In the considered situation the continuous spectrum is absolutely continuous, and explicit formulas for the spectral density and the corresponding Fourier transform are given
Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem

AbstractBy using Schauder's fixed point theorem, we prove some existence results for traveling wavefronts of reaction-diffusion systems with quasimonotonicity reactions. More precisely, we reduce the existence of traveling wavefronts to the existence of an admissible pair of supersolution and subsolution which are easy to construct in practice. Finally, to illustrate our main results, we study the existence of traveling wavefronts for a delayed predator-prey model with diffusion as well as the reaction-diffusion system with the well-known Belousov–Zhabotinskii reaction, and the obtained results improve the existing ones
Degenerate Two-Phase Incompressible Flow I. Existence, Uniqueness and Regularity of a Weak Solution

AbstractThis is the first paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we first show that this system possesses a weak solution under physically reasonable hypotheses on the data. Then we prove that this weak solution is unique. Finally, we establish regularity on the weak solution which is needed in the uniqueness proof. In particular, the Hölder continuity of the saturation in space and time and the Lipschitz continuity of the pressure in space are obtained
Nonexistence of Riemann Solutions and Majda–Pego Instability

AbstractWe study Riemann problems for the shallow water equations. We consider weak self-similar Riemann solutions consisting of constant states, rarefaction waves, and/or jump discontinuities that satisfy the viscous profile entropy criterion, with a positive definite, symmetric viscosity matrix. We prove that for a “generic” symmetric, positive definite viscosity matrix there is an open set of Riemann initial data for which a weak self-similar Riemann solution does not exist. We show that this happens for the hyperbolic initial data that is unstable in the sense studied by Majda and Pego. We prove that such initial data always exist for positive definite, symmetric, nondiagonal viscosity matrices. In the work that follows previous work by the authors (in press, Nonlinear Anal.) we show that in the situations presented in this paper, measure-value solutions exhibiting continuously generated oscillations take place. The results of the present paper provide a new insight into the role of the viscous profile entropy criterion and the Majda–Pego instability in the existence of Riemann solutions for nonlinear conservation laws
On the Local Well Posedness of Quasilinear Schrödinger Equations in Arbitrary Space Dimension

AbstractA class of quasilinear Schrödinger equations is studied which contain strongly singular nonlinearities including derivatives of second order. Such equations have been derived as models of several physical phenomena. The results of this paper apply to the superfluid film equation in fluid mechanics. The local well posedness for smooth solutions of the Cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value. This improves results in the literature which cover the one-dimensional case only. Almost global well posedness results are obtained as well. The proof combines Nash–Moser techniques on implicit function theorems in Fréchet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all Sobolev spaces as basic function space. The nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order Sobolev norms for the linearized problem. Here a certain trace has to vanish in view of apparent integrability conditions. The transformations and the regularity theory are substantially different from the easier one-dimensional case. In particular, continuity estimates are proved for the operator norms of the linear evolution operator in negative Sobolev spaces
On the Stability of Periodic Travelling Waves with Large Spatial Period

AbstractIn many circumstances, a pulse to a partial differential equation (PDE) on the real line is accompanied by periodic wave trains that have arbitrarily large period. It is then interesting to investigate the PDE stability of the periodic wave trains given that the pulse is stable. Using the Evans function, Gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves. In this article, the precise location of these circles is determined. It is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse. As a consequence, periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi-hump pulses are created. That is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations. The proof uses Lyapunov–Schmidt reduction and relies on the existence of exponential dichotomies. The approach is also applicable to periodic waves with large spatial period of elliptic problems on Rn or on unbounded cylinders R×Ω with Ω bounded
On Integrability of Systems of Evolution Equations

AbstractWe prove the conjecture, formulated in [BSW98], that almost all systems in the family[formula]have at most finitely many symmetries by using number theory. We list the nine exceptional cases when the systems do have infinitively many symmetries. For such systems, we give the recursive operators to generate their symmetries. We treat both 1the commutative and the noncommutative (or quantum) cases. This is the first example of a class of equations where such a classification has been possible
Invariant Sets of Descending Flow in Critical Point Theory with Applications to Nonlinear Differential Equations

AbstractProperties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a Banach space are studied. In this way, several critical points can be found by constructing different invariant sets on which the functional is bounded below. Under suitable conditions, the existence of at least four critical points of a functional is proved, each critical point being in a certain invariant set. The theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations. In variant cases, at least four solutions are obtained for these equations
Uniqueness of Classical and Nonclassical Solutions for Nonlinear Hyperbolic Systems

AbstractIn this paper we establish a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one-space dimension. First of all we introduce a new notion of admissible solutions based on prescribed sets of admissible discontinuitiesΦ and admissible speedsψ. Our definition unifies in a single framework the various notions of entropy solutions known for hyperbolic systems of conservation laws, as well as for systems in nonconservative form. For instance, it covers the nonclassical (undercompressive) shock waves generated by a vanishing diffusion-dispersion regularization and characterized via a kinetic relation. It also covers Dal Maso, LeFloch, and Murat's definition of weak solutions of nonconservative systems. Under certain natural assumptions on the prescribed sets Φ and ψ and assuming the existence of a L1-continuous semi-group of admissible solutions, we prove that, for each Cauchy datum at t=0, there exists at most one admissible solution to the Cauchy problem depending L1-continuously upon the initial data. In particular, our result shows the uniqueness of the L1-continuous semi-group of admissible solutions. In short, this paper proves that supplementing a hyperbolic system with the “dynamics” of elementary discontinuities characterizes at most one L1-continuous and admissible solution
A System of Resource-Based Growth Models with Two Resources in the Unstirred Chemostat

AbstractModels of single-species growth in the unstirred chemostat on two growth-limiting, nonreproducing resources are considered. For the case of two complementary resources, the existence and uniqueness of a positive steady-state solution is established. It is also proved that the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values. For the case of two substitutable resources, the existence of a positive steady-state solution is determined for a range of the parameter (m, n). Techniques include the maximum principle, monotone method and global bifurcation theory. The longtime behavior of the corresponding limiting system is given for a range of (m, n). In the special case of m=n, the uniqueness and global attractivity of the positive steady-state solution of the original system is established
Convergence to Traveling Waves in Two Model Systems Related to the Dynamics of Liquid–Vapor Phase Changes

AbstractTwo prototype systems arising from an earlier model for the dynamics of liquid–vapor phase transitions of retrograde fluids in shock tubes are studied. For the first system, the existence of traveling waves is established. The stability of some traveling waves is proved under some conditions if the traveling wave for the reaction-diffusion part of the system is stable. This shows that the admissibility of traveling waves for the original model cannot be determined by stability consideration alone. This leads to the second prototype system for the original model in which the effect of initiation of nuclei is included. For the second system, the convergence to traveling waves is proved for some initial value problems. This result shows the initiation term has a sizable effect in the long time on the speed of the phase boundary
On a Planar System Modelling a Neuron Network with Memory

AbstractWe consider a delay differential equation modelling a network of two neurons with memory. The situations without self-connections and two delays, and with self-connections and two equal delays, are addressed in terms of local stability and bifurcation analysis. In the first case, the dynamical behavior is studied by taking one of the delays as the bifurcating parameter. Namely, the Hopf bifurcation, whose existence was proven in former works, occurs as the delay crosses some critical values and is completely described. Conditions ensuring the stability of the periodic cycles are given. When self-connections are present, Hopf and pitchfork bifurcations co-exist and the equation describing the flow on the center manifold is derived
Homoclinic Bifurcation in an SIQR Model for Childhood Diseases

AbstractWe consider a system of ODEs which describes the transmission dynamics of childhood diseases. A center manifold reduction at a bifurcation point has the normal form x′=y, y′=axy+bx2y+O(4), indicating a bifurcation of codimension greater than two. A three-parameter unfolding of the normal form is studied to capture possible complex dynamics of the original system which is subjected to certain constraints on the state space due to biological considerations. It is shown that the perturbed system produces homoclinic bifurcation
Zero Reaction Limit for Hyperbolic Conservation Laws with Source Terms

AbstractIn this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source term∂tu+∂xf(u)=1εu(1−u2).For the Cauchy problem to the above equation, we prove that as ε→0, its solution converges to piecewise constant (±1) solution, where the two constants are the two stable local equilibria. The constants are separated by either shocks that travel with speed 12(f(1)−f(−1)), as determined by the Rankine-Hugoniot jump condition, or a non-shock discontinuity that moves with speed f′(0), where 0 is the unstable equilibrium. Our analytic tool is the method of generalized characteristics. Similar results for more general source term 1εg(u), having finitely many simple zeros and satisfying ug(u)<0 for large |u|, are also given
Dynamics of Differential Equations on Invariant Manifolds

AbstractThe simplification resulting from reduction of dimension involved in the study of invariant manifolds of differential equations is often difficult to achieve in practice. Appropriate coordinate systems are difficult to find or are essentially local in nature thus complicating analysis of global dynamics. This paper develops an approach which avoids the selection of coordinate systems on the manifold. Conditions are given in terms compound linear differential equations for the stability of equilibria and periodic orbits. Global results include criteria for the nonexistence of periodic orbits and a discussion of the nature of limit sets. As an application, a global stability criterion is established for the endemic equilibrium in an epidemiological model
Generalized Rankine–Hugoniot Condition and Shock Solutions for Quasilinear Hyperbolic Systems

AbstractFor a quasilinear hyperbolic system, we use the method of vanishing viscosity to construct shock solutions. The solution consists of two regular regions separated by a free boundary (shock). We use Melnikov's integral to obtain a system of differential/algebraic equations that governs the motion of the shock. For Lax shocks in conservation laws, these equations are equivalent to the Rankine–Hugoniot condition. For under compressive shocks in conservation laws, or shocks in non-conservation systems, the Melnikov-type integral obtained in this paper generalizes the Rankine–Hugoniot condition. Under some generic conditions, we show that the initial value problem of shock solutions can be solved as a free boundary problem by the method of characteristics
Center Manifolds for Invariant Sets

AbstractWe derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in Rn. By applying the Hadamard graph transform technique, it is shown that, associated to a natural dynamical characteristic of the linearized flow along the invariant set, there exists an invariant manifold (called a center manifold) of the invariant set which contains every locally bounded solution (in particular, contains the invariant set) and is persistent under small perturbations
Surface Nucleation of Superconductivity in 3-Dimensions

AbstractIn this paper we study the surface nucleation of superconductivity and estimate the value of the upper critical field HC3 for superconductors occupying arbitrary bounded smooth domains in R3. We show that HC3≃κ/β0, the ratio of the Ginzburg–Landau parameter κ and the first eigenvalue β0 of the Schrödinger operator with unit magnetic field on the half plane. When the applied magnetic field is spacially homogeneous and close to HC3, a superconducting layer nucleates on a portion of the surface at which the applied field is tangential to the surface. Nucleation under non-homogeneous applied fields is also discussed
The Principal Spectrum for Linear Nonautonomous Parabolic PDEs of Second Order: Basic Properties

AbstractWe put forward a theory extending the notion of principal eigenfunction and principal eigenvalue to the case of linear nonautonomous parabolic PDEs of second orderut=∑i, j=1Naij(t, x)∂2u∂xi∂xj+∑i=1Nai(t, x)∂u∂xi+a0(t, x)u,  x∈Ω,on a bounded domain Ω⊂RN, with Dirichlet or Robin boundary conditions. A canonically defined one-dimensional subbundle S (corresponding to the solutions that are globally defined and of the same sign) serves as an analog of principal eigenfunction. The principal spectrum is defined to be the dynamical (Sacker–Sell) spectrum of the linear skew-product flow on S. Characterizations of principal spectrum in terms of (logarithmic) growth rates of positive solutions are given. Finally, monotonicity of the principal spectrum with respect to zero order terms is proved
Dynamics of a Quadratic Map in Two Complex Variables

AbstractWe study the map Ψ:C2→C2 defined by Ψ(w, z)=(z, z+w2) and the associated collection of sequences {zj} satisfying the recurrence zj+2=zj+1+z2j. Iteration of Ψ is equivalent to the study of such sequences. We analyze growth rates of the sequences, positive sequences, periodic and asymptotically periodic sequences and establish the existence of doubly infinite homoclinic sequences, non-zero sequences whose limits as j→±∞ are 0. We investigate some associated functional equations, f (x+2)=f(x+1)+f(x)2 and L(L(x))=L(x)+x2
The Non-Holonomic Mechanics

AbstractTwo approaches for the study of mechanical systems with non-holonomic constraints are presented: d'Alembertian mechanics and variational (vakonomic) mechanics. The first one is equivalent to the d'Alembert principle and the second comes from a variational principle. In order to develop the two approaches, d'Alembertian and vakonomic trajectories are introduced. A generalized version of the classical Liouville theorem for the conservation of volume is proved in the context of d'Alembertian mechanics. A characterization for the notions of regular and singular curves is presented. The flow corresponding to the regular vakonomic trajectories is Hamiltonian