pinecone/core-2020-05-10-deduplication huggingface dataset:
Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results

AbstractFor the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray–Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence
On Differential Systems Describing Surfaces of Constant Curvature

AbstractThe geometric notion of a differential system describing surfaces of constant nonzero Gaussian curvature is introduced. The nonlinear Schrödinger equation (NLS) with κ=1 and −1 is shown to describe a family of spherical surfaces (s.s.) and pseudospherical surfaces (p.s.s.), respectively. The Schrödinger flow of maps into S2 (the HF model) and its generalized version, the Landau–Lifschitz equation, are shown to describe spherical surfaces. The Schrödinger flow of maps into H2 (the M-HF model) provides another example of a system describing pseudo-spherical surfaces. New differential systems describing surfaces of nonzero constant Gaussian curvature are obtained. Furthermore, we give a characterization of evolution systems which describe surfaces of nonzero constant Gaussian curvature. In particular, we determine all differential systems of typeut=−vxx+H11(u,v)ux+H12(u,v)vx+H13(u,v),vt=−uxx+H21(u,v)ux+H22(u,v)vx+H23(u,v), which describe η-pseudospherical or η-spherical surfaces. As an application, we obtain four-parameter family of such systems for a complex-valued function q=u+iv given by iqt+qxx±iγ(∣q∣2q)x−iαqx±σ∣q∣2q−βq=0, where σ⩾0 if γ=0. Particular cases of this family, obtained by the vanishing of the parameters, are the linear equations, the NLS equation, the derivative nonlinear Schrödinger equation (DNLS) and the mixed NLS–DNLS equation
Obstructions to Existence in Fast-Diffusion Equations

AbstractThe study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory. Thus, in the sub-field of very fast diffusion it is known that the Cauchy problem can be ill-posed, either because of non-uniqueness, or because of non-existence of solutions with small data. The equations we consider take the general form ut=(D(u,ux)ux)x or its several-dimension analogue. Fast diffusion means that D→∞ at some values of the arguments, typically as u→0 or ux→0. Here, we describe two different types of non-existence phenomena. Some fast-diffusion equations with very singular D do not allow for solutions with sign changes, while other equations admit only monotone solutions, no oscillations being allowed. The examples we give for both types of anomaly are closely related. The most typical examples are vt=(vx/∣v∣)x and ut=uxx/∣ux∣. For these equations, we investigate what happens to the Cauchy problem when we take incompatible initial data and perform a standard regularization. It is shown that the limit gives rise to an initial layer where the data become admissible (positive or monotone, respectively), followed by a standard evolution for all t>0, once the obstruction has been removed
On the Dynamics of the Classical Transverse Field Ising Chain

AbstractWe investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions
Self-Similar Solutions to a Parabolic System Modeling Chemotaxis

AbstractWe study the forward self-similar solutions to a parabolic system modeling chemotaxisut=∇·(∇u−u∇v),τvt=Δv+u in the whole space R2, where τ is a positive constant. Using the Liouville-type result and the method of moving planes, it is proved that self-similar solutions (u,v) must be radially symmetric about the origin. Then the structure of the set of self-similar solutions is investigated. As a consequence, it is shown that there exists a threshold in ∫R2u for the existence of self-similar solutions. In particular, for 0<τ⩽1/2, there exists a self-similar solution (u,v) if and only if ∫R2u<8π
The Balance between Nonlinear Inwards and Outwards Boundary Flux for a Nonlinear Heat Equation

AbstractWe study nonnegative solutions ofut=(um)xx(x,t)∈(0,L)×(0,T),−(um)x(0,t)=up(0,t),t∈(0,T),(um)x(L,t)=−λuq(L,t),t∈(0,T),u(x,0)=u0(x),x∈(0,L), where m, p, q, λ and L are positive parameters. For different values of the parameters three situations may occur: (1) all solutions of this problem exist for all t>0; (2) for certain initial data functions the solution exists for all t>0 while for others the solution blows up as t↗T for some finite T; (3) excepting the trivial solution when u0≡0, all solutions blow up as t↗T for some finite T. We identify in terms of the parameters which of them actually happens. For solutions which blow up we find the blow-up rate and the blow-up set
SU(3) Chern–Simons Vortex Theory and Toda Systems

AbstractMotivated by the study of the asymptotic properties of “non-topological” condensates in the non-abelian Chern–Simons vortex theory (see [26]), we analyze the SU(3) Toda system: (P)λ1,λ2−Δzj=λj(e∑2i=1kijzI/∫Me∑i=12kijzi−1)on M,∫Mzj=0j=1,2, where M=R2/Z2, K=(kij)=2−1−12 is the SU(3) Cartan matrix and λj are positive parameters. We study the variational problem associated with the system (P)λ1,λ2 in a range of parameters, where the trivial solution is a strict local minimum and the corresponding Sobolev-type inequality fails to apply. In this situation, a lack of compactness may occur due to concentration phenomena. Nonetheless, we are able to establish the existence of a non-trivial solution for (P)λ1,λ2 which is not a minimizer
Existence and a Priori Estimates for Positive Solutions of p-Laplace Systems

AbstractWe use continuation and moving hyperplane methods to prove some existence and a priori estimates for p-Laplace systems of the form−Δp1u=f(∣v∣) in Ω,u=0 on ∂Ω,−Δp2v=g(∣u∣) in Ω,v=0 on ∂Ω, where 1<p1,p2<N, Ω⊂RN is bounded and convex, and f,g : R→R+ are nondecreasing locally Lipschitz continuous functions satisfyingC1∣s∣q1⩽f(s)⩽C2∣s∣q1,D1∣s∣q2⩽g(s)⩽D2∣s∣q2 ∀s∈R+ for some positive constants C1,C2,D1,D2 and q1q2(p1−1)(p2−1). We extend results obtained in Azizieh and Clément (J. Differential Equations, 179 (2002), 213–245) where the case of a single equation was considered
The Total Variation Flow in RN

AbstractIn this paper, we study the minimizing total variation flow ut=div(Du/∣Du∣) in RN for initial data u0 in Lloc1(RN), proving an existence and uniqueness result. Then we characterize all bounded sets Ω of finite perimeter in R2 which evolve without distortion of the boundary. In that case, u0=χΩ evolves as u(t,x)=(1−λΩt)+χΩ, where χΩ is the characteristic function of Ω, λΩ≔P(Ω)/∣Ω∣, and P(Ω) denotes the perimeter of Ω. We give examples of such sets. The solutions are such that v≔λΩχΩ solves the eigenvalue problem −divDv∣Dv∣=v. We construct other explicit solutions of this problem. As an application, we construct explicit solutions of the denoising problem in image processing
Discreteness of the Spectrum of Second-Order Differential Operators and Associated Embedding Theorems

AbstractNecessary and sufficient conditions and also simple sufficient conditions are given for the self-adjoint operators associated with the second-order linear differential expressionτ(y)=1w(−(py′)′+qy) on [a,b) to have discrete spectrum. Here the coefficients of τ are non-negative and satisfy minimal smoothness conditions. These results follow from compact embedding theorems from a weighted one-dimensional Sobolev space with norm∫ab(p∣f′∣r+q∣f∣r))1/r into a weighted Banach space with norm(∫abw∣f∣s)1/s
Existence and Asymptotic Stability of Traveling Waves of Discrete Quasilinear Monostable Equations

AbstractWe study the existence and asymptotic stability of traveling waves tout=[g(u+1)+g(u−1)−2g(u)]+f(u) on R×(0,∞) , where u=u(x,t), u±1=u(x±1,t), g=dup (d>0, p⩾1) and f=u−u2. We show that there exists c>0 such that for each wave speed c>c, there is a traveling wave U∈C1(R), i.e., a solution of the form u=U(x−ct). The traveling wave has the property that U(−∞)=1, U′<0 on R, and limξ→∞U(ξ)eλξ=1, where λ=Λ1(c) is the smallest solution to cλ=f′(0)+g′(0)[eλ+e−λ−2]. We also show that the traveling wave is globally asymptotically stable in the sense that if an initial value u(·,0)∈C(R→[0,1]) satisfies lim infx→−∞u(x,0)>0 and limx→∞u(x,0)eλx=1 for some λ∈(0,Λ1(c)), then limt→∞ supR ∣u(·+ct,t)/U(·)−1∣=0 where (c,U) is the traveling wave with speed c=C(λ)={f′(0)+g′(0)[eλ+e−λ−2]}/λ, the inverse of λ=Λ1(c)
Spatial Heterogeneity of Resources Versus Lotka-Volterra Dynamics

AbstractThe problem is motivated by a consideration of two phenotypes of a species in a strongly heterogenous spatial environment. The phenotypes vary in their diffusion rates and interspecific interactions. The aim is to investigate the relative strengths of the diffusion and reaction effects. The problem is thus of competing species type, but there are many questions which arise which are not covered by standard theory. We investigate, in particular, the stability of the equilibria and the existence of coexistence solutions with emphasis on cases where the spatial variation of the environment becomes large. We discuss briefly the implications of our results for the principle of competitive exclusions and for the question of the evolution of diffusion discussed in Dockery et al. (J. Math. Biol.37 (1998), 61–83)
The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula

AbstractFor a bounded domain Ω⊂R2, we establish a concentration-compactness result for the following class of “singular” Liouville equations:−Δu=eu−4π∑j=1mαjδpj in Ω where pj∈Ω, αj>0 and δpj denotes the Dirac measure with pole at point pj, j=1,…,m. Our result extends Brezis–Merle's theorem (Comm. Partial Differential Equations16 (1991) 1223–1253) concerning solution sequences for the “regular” Liouville equation, where the Dirac measures are replaced by Lp(Ω)-data p>1. In some particular case, we also derive a mass-quantization principle in the same spirit of Li–Shafrir (Indiana Univ. Math. J.43 (1994) 1255–1270). Our analysis was motivated by the study of the Bogomol'nyi equations arising in several self-dual gauge field theories of interest in theoretical physics, such as the Chern–Simons theory (“Self-dual Chern–Simons Theories” Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin, 1995) and the Electroweak theory (“Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions,” World Scientific, Singapore)
Weak Entropy Boundary Conditions for Isentropic Gas Dynamics via Kinetic Relaxation

AbstractWe consider a kinetic BGK model relaxing to isentropic gas dynamics previously introduced by the authors, but with Dirichlet boundary condition on the incoming velocities. We pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain, and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes. Our method is very general and could be applied to any entropy satisfying BGK model as soon as we have strong compactness of the macroscopic variables inside the domain
On a Class of Semilinear Elliptic Equations in Rn

AbstractWe establish that for n⩾3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses separated positive entire solutions of infinite multiplicity, provided that a locally Hölder continuous function K⩾0 in Rn\{0}, satisfies K(x)=O(∣x∣σ) at x=0 for some σ>−2, and K(x)=c∣x∣−2+O(∣x∣−n[log∣x∣]q) near ∞ for some constants c>0 and q>0. In the radial case K(x)=∣x∣l1+∣x∣τ with l>−2 and τ⩾0, or K(x)=∣x∣λ−2(1+∣x∣2)λ/2 with λ>0, we investigate separation phenomena of positive radial solutions, and show that if n and p are large enough, the equation possesses a positive radial solution with initial value α at 0 for each α>0 and a unique positive radial singular solution among which any two solutions do not intersect
Asymptotic Smoothing and the Global Attractor of a Weakly Damped KdV Equation on the Real Line

AbstractThe existence of the global attractor of a weakly damped, forced Korteweg–de Vries equation in the phase space L2(R) is proved. An optimal asymptotic smoothing effect of the equation is also shown, namely, that for forces in L2(R), the global attractor in the phase space L2(R) is actually a compact set in H3(R). The energy equation method is used in conjunction with a suitable splitting of the solutions; the dispersive regularization properties of the equation in the context of Bourgain spaces are extensively exploited
A Shooting Approach to Layers and Chaos in a Forced Duffing Equation

AbstractWe study equilibrium solutions for the problemut=ε2uxx−u3+λu+cos x,ux(0, t)=ux(1, t)=0. Using a shooting method we find solutions for all nonzero ε. For small ε we add to the solutions found by previous authors, especially Angenent, Mallet-Paret and Peletier, and Hale and Sakamoto, and also give new elementary ode proofs of their results. Among the new results is the existence of internal layer-type solutions. Considering the ode satisfied by equilibria, but on an infinite interval, we obtain chaos results for λ⩾λ0=322/3 and 0<ε⩽14. We also consider the problem of bifurcation of solutions as λ increases
The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

AbstractIn this paper, we study the Fučik spectrum of the problem: (*) ẍ+(λ++q+(t))x++(λ−+q−(t))x−=0 with the 2π-periodic boundary condition, where q±(t) are 2π-periodic. After introducing a rotation number function ρ(λ+, λ−) for (*), we prove using the Hamiltonian structure and the positive homogeneity of (*) that for any positive integer n, the two boundary curves of the domain ρ−1(n/2) in the (λ+, λ−)-plane are Fučik curves of (*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fučik spectrum with the Dirichlet condition. In particular, it remains open if the Fučik spectrum of (*) is composed of only these curves
Uniform Boundedness and Convergence of Solutions to Cross-Diffusion Systems

AbstractUniform boundedness and convergence of global solutions are proved for cross-diffusion systems in population dynamics. Gagliardo–Nirenberg type inequalities are used in the estimates of solutions in order to establish W21-bounds uniform in time. In each step of estimates the contribution of the diffusion coefficients are emphasized, and it is concluded that the uniform bound remains independent of the growth of the diffusion coefficient in the system. Hence convergence of solutions are established for systems with large diffusion coefficients
Linking–Preserving Perturbations of Symmetric Functionals

AbstractWe consider paths of functionals starting with one which is invariant under the action of an arbitrary group of symmetries. We give conditions for the existence of an unbounded sequence of critical values of the non-symmetric functional at the end of the path in terms of the growth of the critical values of the symmetric one. We apply this to obtain a multiplicity result for a system of elliptic equations whose symmetries are perturbed by a linear term and a non-homogeneous boundary condition
The Cauchy Problem and Stability of Solitary-Wave Solutions for RLW–KP-Type Equations

AbstractThe Kadomtsev–Petviashvilli (KP) equation,(ut+ux+uux+uxxx)x+εuyy=0, (*) arises in various contexts where nonlinear dispersive waves propagate principally along the x-axis, but with weak dispersive effects being felt in the direction parallel to the y-axis perpendicular to the main direction of propagation. We propose and analyze here a class of evolution equations of the form (ut+ux+upux+Lut)x+εuyy=0, (**) which provides an alternative to Eq.(*) in the same way the regularized long-wave equation is related to the classical Korteweg–de Vries (KdV) equation. The operator L is a pseudo-differential operator in the x-variable, p⩾1 is an integer and ε=±1. After discussing the underlying motivation for the class (**), a local well-posedness theory for the initial-value problem is developed. With assumptions on L and p that include conditions appertaining to models of interesting physical phenomenon, the solutions defined locally in time t are shown to be smoothly extendable to the entire time-axis. In the particularly interesting case where L=−∂x2 and ε=−1, (*) possesses travelling-wave solutions u(x,y, t)=πc(x−ct,y) provided c>1 and 0<p<4. It is shown here that these solitary waves are stable for 0<p<43 and c>1 and for 43<p<4 if c>(4p)/(4+p). The paper concludes with commentary on extensions of the present theory to more than two space dimensions
A New Criterion for Controlling the Number of Limit Cycles of Some Generalized Liénard Equations

AbstractWe consider a class of planar differential equations which include the Liénard differential equations. By applying the Bendixson–Dulac Criterion for ℓ-connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign. As an application, this method is used to give a sharp upper bound for the number of limit cycles of some Liénard differential equations. In particular, we present a polynomial Liénard system with exactly three limit cycles
On the Number and Structure of Solutions for a Fredholm Alternative with the p-Laplacian

AbstractWe investigate the existence and multiplicity of weak solutions u∈W01,p(Ω) to the degenerate quasilinear Dirichlet boundary value problem(P) −Δpu=λ1∣u∣p−2u+fT(x)+ζ·ϕ1(x) inΩu=0 on ∂Ω, where ζ∈R is a parameter. It is assumed that 1<p<∞, p≠2, and Ω is a bounded domain in RN. The number λ1 stands for the first (smallest) eigenvalue of the positive p-Laplacian − Δp, where Δpu≡div(∣ ∇u∣p−2∇u). The eigenvalue λ1 being simple, let ϕ1 denote the eigenfunction associated with λ1. Furthermore, fT∈L∞(Ω) is a given function which is assumed to be L2-orthogonal to ϕ1 and fT≢0 in Ω. We show the existence of a solution for problem (P) precisely when the parameter ζ satisfies ζ*⩽ζ⩽ζ*, for some numbers −∞<ζ*<0<ζ*<∞ depending on fT. Otherwise, nonexistence occurs. Moreover, we show that problem (P) possesses at least two distinct solutions provided ζ#<ζ<ζ# and ζ≠0, for some numbers ζ# and ζ# such that ζ*⩽ζ#<0<ζ#⩽ζ*. Finally, given any δ≷0, we show that the set of all weak solutions to problem (P) is bounded in C1(Ω) uniformly for ∣ζ∣⩾δ and for ζ=0 as well. Precise asymptotic behavior (blow-up) of every solution is given as ζ→0 (ζ≠0) using the linearization of equation (P) about ϕ1
Asymptotic Stability of Traveling Wave Solutions for Perturbations with Algebraic Decay

AbstractFor a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations
Reading and spelling acquisition in French: the role of phonological mediation and orthographic factors.

International audienceThe objective of this research was to study the development of reading and spelling in French. The two main hypotheses were that (1) phonological mediation is the primary process in the acquisition of these skills and that (2) the use of phonological mediation may allow the construction of the orthographic lexicon. In January and June, first graders (n = 57) were required to read and spell items designed to assess the variables of regularity, graphemic complexity, frequency, lexicality and analogy. The findings of the January session partially corroborated the first hypothesis as a regularity effect, but no frequency effect and no word superiority, were found both in reading and spelling. The main contradictory finding was the presence, in early reading only, of a facilitative effect of analogy. The changes in the frequency and the lexicality effects between the two sessions in reading and in spelling indicated that the children were able to rapidly construct an orthographic lexicon. However, this procedure did not entirely replace phonological mediation since a regularity effect and regularization errors were observed and increased between sessions. The second hypothesis was supported as relationships were found to exist between early phonological skills and subsequent orthographic skills. Finally, we observed that French children were using graphemes (not only letters), in the early stage of reading, and, to a lesser extent, in the early stage of spelling. The findings are discussed in the context of developmental models of reading and spelling
The Development of Working Memory: Exploring the Complementarity of Two Models

Author version made available in accordance with the publisher's policy for non-mandated open access submission.  Under Elsevier's copyright, non-mandated authors are permitted to make work available in an institutional repository.The aim of the paper was to further explore the complementarity of the working memory models postulated by Pascual-Leone and Baddeley. Five-, six-, eight- and nine-year-old children were assessed on two working memory tasks, which have frequently been used within the respective streams of research: the Mr. Peanut task and the Corsi blocks task. Results indicated a developmental increase in spatial short-term memory for both tasks. Concurrent spatial suppression reduced performance on the two tasks in all four age groups. By contrast, articulatory suppression only interfered with recall on the Mr. Peanut task, and only in the older children. The two models were shown to make their own specific contribution to the interpretation of the data, attesting to their complementarity. Pascual-Leone’s theory offered a clear explanation of the results concerning the central aspects of working memory, i.e. the stepwise age-related increase in performance, whereas Baddeley’s model provided a convincing account of the findings regarding the peripheral phonological and visuo-spatial components, i.e. the effects of articulatory and spatial suppression.\ud
Key Words: short-term memory; spatial memory; information processing capacity; memory development; working memory models; rehearsal strategies; cognitive development