pinecone/core-2020-05-10-deduplication huggingface dataset:
A Semilinear Parabolic System for Migration and Selection in Population Genetics

AbstractThe semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The selection coefficients depend on position; the drift and diffusion coefficients may do so. For two alleles (the scalar case), the global analysis of D. Henry (1981, “Geometric Theory of Semilinear Parabolic Equations,” Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin) is extended from homogeneous, isotropic migration (corresponding to the Laplacian) to arbitrary migration (corresponding to an arbitrary elliptic operator). For multiple alleles, sufficient conditions are given for the global loss of an allele that is nowhere the fittest. In the case of no dominance with at least one change in the direction of selection, sufficient conditions are established for global convergence to a stable equilibrium with all the intermediate alleles absent and one or two extreme alleles present. Sufficient conditions on the migration pattern for casting the elliptic operator into variational form are proved; in this case, the above results become more explicit
Large Time Behavior via the Method of ℓ-Trajectories

AbstractThe method of ℓ-trajectories is presented in a general setting as an alternative approach to the study of the large-time behavior of nonlinear evolutionary systems. It can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear. Here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics. In both cases we prove the existence of a finite-dimensional global attractor and the existence of an exponential attractor
Solutions of Hamilton–Jacobi Equations and Scalar Conservation Laws with Discontinuous Space–Time Dependence

AbstractWe establish a unique stable solution to the Hamilton–Jacobi equation ut+H(K(x,t),ux)=0, x∈(−∞,∞), t∈[0,∞) with Lipschitz initial condition, where K(x,t) is allowed to be discontinuous in the (x,t) plane along a finite number of (possibly intersecting) curves parameterized by t. We assume that for fixed k, H(k,p) is convex in p and limp→±∞∣H(k,p)p∣=∞. The solution is determined by showing that if K is made smooth by convolving K in the x direction with the standard mollifier, then the control theory representation of the viscosity solution to the resulting Hamilton–Jacobi equation must converge uniformly as the mollification decreases to a Lipschitz continuous solution with an explicit control theory representation. This also defines the unique stable solution to the corresponding scalar conservation law ut+(f(K(x,t),u))x=0, x∈(−∞,∞), t∈[0,∞) with K discontinuous
Linear Stability of the Elliptic Lagrangian Triangle Solutions in the Three-Body Problem

AbstractThis paper concerns the linear stability of the well-known periodic orbits of Lagrange in the three-body problem. Given any three masses, there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic Kepler orbit. Reductions are performed to derive equations which determine the linear stability of the periodic solutions. These equations depend on two parameters – the eccentricity e of the orbit and the mass parameter β=27(m1m2+m1m3+m2m3)/(m1+m2+m3)2. A combination of numerical and analytic methods is used to find the regions of stability in the βe-plane. In particular, using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not
Heteroclinic and Periodic Cycles in a Perturbed Convection Model

AbstractVivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast – slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) – solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov–Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity
Periodic Solutions of the Forced Pendulum: Exchange of Stability and Bifurcations

AbstractWe study the T-periodic solutions of the forced pendulum equation u″+cu′+a sin(u)=λf(t), where f satisfies f(t+T2)=−f(t). We prove that this equation always has at least two geometrically distinct T-periodic solutions u0 and u1. We then investigate the dependence of the set of T-periodic solutions on the forcing strength λ. We prove that under some restriction on the parameters a,c, the periodic solutions found before can be smoothly parameterized by λ, and that there are some λ-intervals for which u0(λ), u1(λ) are the only T-periodic solutions up to geometrical equivalence, but there are other λ-intervals in which additional T-periodic solutions bifurcate off the branches u0(λ), u1(λ). We characterize the global structure of the bifurcating branches. Related to this bifurcation phenomenon is the phenomenon of ‘exchange of stability’ – in some λ-intervals u0(λ) is dynamically stable and u1(λ) is unstable, while in other λ-intervals the reverse is true, a phenomenon which has important consequences for the dynamics of the forced pendulum, as we show by both theoretical analysis and numerical simulation
Infinitely Many Critical Points of Non-Differentiable Functions and Applications to a Neumann-Type Problem Involving the p-Laplacian

AbstractFor a family of functionals in a Banach space, which are possibly non-smooth and depend also on a positive real parameter, the existence of a sequence of critical points (according to Motreanu and Panagiotopoulos (“Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,” Nonconvex Optimization Applications, Vol. 29, Kluwer, Dordrecht, 1998, Chap. 3)) is established by mainly adapting a new technique due to Ricceri (2000, J. Comput. Appl. Math.113, 401–410). Two applications are then presented. Both of them treat the Neumann problem for an elliptic variational–hemivariational inequality with p-Laplacian
The Falkner–Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

AbstractThe Falkner–Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336–394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur
Doubly Nonlinear Degenerate Parabolic Systems with Coupled Nonlinear Boundary Conditions

AbstractIn this paper, we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions. We first prove a local existence result by the regularization method. Next, we construct a weak comparison principle. Then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained
Energy Levels of Steady States for Thin-Film-Type Equations

AbstractWe study the phase space of the evolution equationht=−(f(h)hxxx)x−(g(h)hx)x by means of a dissipated energy. Here h(x,t)⩾0, and at h=0 the coefficient functions f>0 and g can either degenerate to 0, or blow up to ∞, or tend to a nonzero constant. We first show that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, if (g/ f)″⩾0 or if the perturbations are allowed to have period longer than that of the steady-state. For power-law coefficients (f (y)= yn and g(y)=Bym for some B>0) we analytically determine the relative energy levels of distinct steady-states. For example, with m−n∈[1,2) and for suitable choices of the period and mean value, we find three fundamentally different steady-states. The first is a constant steady-state that is stable and is a local minimum of the energy. The second is a positive periodic steady-state that is linearly unstable and has higher energy than the constant steady-state; it is a saddle point for the energy. The third is a periodic collection of ‘droplet’ (compactly supported) steady-states having lower energy than either the positive steady-state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation
Traveling Wave Solutions for Planar Lattice Differential Systems with Applications to Neural Networks

AbstractWe obtain some existence results for traveling wave fronts and slowly oscillatory spatially periodic traveling waves of planar lattice differential systems with delay. Our approach is via Schauder's fixed-point theorem for the existence of traveling wave fronts and via S1-degree and equivarant bifurcation theory for the existence of periodic traveling waves. As examples, the obtained abstract results will be applied to a model arising from neural networks and explicit conditions for traveling wave fronts and global continuation of periodic waves will be obtained
Periodic Problems for Strongly Nonlinear Second-Order Differential Inclusions

AbstractIn this paper, we study periodic problems for second-order differential inclusions in RN with a maximal monotone term. The nonlinear differential operator is not necessarily homogeneous, does not obey any growth condition and incorporates as a special case the one-dimensional p-Laplacian. Using techniques from multivalued analysis and the theory of operators of monotone type, we prove the existence of solutions for both the “convex” and “nonconvex” problems, when the maximal monotone term A is defined everywhere and when it is not defined everywhere (case of variational inequalities)
Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

AbstractA free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated. The existence, uniqueness, and regularity of global solutions are established with large initial data in H1. It is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. An existence theorem of global solutions with large discontinuous initial data is also established
The Boundary Behavior of Heat Kernels of Dirichlet Laplacians

AbstractWe prove the following complete and qualitatively sharp description of heat kernels G of Dirichlet Laplacians on bounded C1,1 domains D. There exist positive constants c1,c2 and T>0 depending on D such that, for ρ(x)=dist(x,∂D), ρ(x)ρ(y)t∧1c1tn/2e−c2∣x−y∣2/t⩽G(x,t;y,0)⩽ρ(x)ρ(y)t∧11c1tn/2e−∣x−y∣2/(c2t) for all x,y∈D and 0<t⩽T. The upper bound is well known since the 1980s (E.B. Davies, J. Funct. Anal. 71 (1987), 88–103) however, the existence of the lower bound had been an open question since then. (Bounds when t>T are known.) Bounds when D is unbounded are also given
Mountain Pass Solutions for a Double-Well Energy

AbstractWe establish the existence of a mountain pass solution for a variational integral involving a quasiconvex function with a double-well structure in the geometrically linear elasticity setting. We show that under small dead-load perturbations, the Neumann boundary value problem has at least three solutions, a global minimizer, a local minimizer and a mountain pass solution. We show that our variational integral satisfies a Weak Palais–Smale condition (WPS) hence the mountain pass lemma applies
A Sequence of Order Relations: Encoding Heteroclinic Connections in Scalar Parabolic PDE

AbstractWe address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations TYPE="DISPLAY">ut=uxx+f(x,u,ux), 0<x<1 on a bounded interval with Neumann conditions. Introducing a sequence of order relations, we prove a new and simple criterion for the existence of heteroclinic connections, using only information about nodal properties of solutions to the stationary ODE problem. This result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria, given by their order at the two boundaries of the interval
Quasiperiodic Motions in the Planar Three-Body Problem

AbstractIn the direct product of the phase and parameter spaces, we define the perturbing region, where the Hamiltonian of the planar three-body problem is Ck-close to the dynamically degenerate Hamiltonian of two uncoupled two-body problems. In this region, the secular systems are the normal forms that one gets by trying to eliminate the mean anomalies from the perturbing function. They are Pöschel-integrable on a transversally Cantor set. This construction is the starting point for proving the existence of and describing several new families of periodic or quasiperiodic orbits: short periodic orbits associated to some secular singularities, which generalize Poincaré's periodic orbits of the second kind (“Les Méthodes Nouvelles de la Mécanique Céleste,” Vol. 1, Gauthier-Villars, Paris, 1892–1899); quasiperiodic motions with three (resp. two) frequencies in a rotating frame of reference, which generalize Arnold's solutions (Russian Math. Surveys18 (1963), 85–191) (resp. Lieberman's solutions; Celestial Mech.3 (1971), 408–426); and three-frequency quasiperiodic motions along which the two inner bodies get arbitrarily close to one another an infinite number of times, generalizing the Chenciner–Llibre's invariant “punctured tori” (Ergodic Theory Dynam. Systems8 (1988), 63–72). The proof relies on a sophisticated version of KAM theorem, which itself is proved using a normal form theorem of Herman (“Démonstration d'un Théorème de V.I. Arnold,” Séminaire de Systèmes Dynamiques and Manuscripts, 1998)
Homogenization in L∞

AbstractHomogenization of deterministic control problems with L∞ running cost is studied by viscosity solutions techniques. It is proved that the value function of an L∞ problem in a medium with a periodic micro-structure converges uniformly on the compact sets to the value function of the homogenized problem as the period shrinks to 0. Our main convergence result extends that of Ishii (Stochastic Analysis, control, optimization and applications, pp. 305–324, Birkhäuser Boston, Boston, MA, 1999.) to the case of a discontinuous Hamiltonian. The cell problem is solved, but, as non-uniqueness occurs, the effective Hamiltonian must be selected in a careful way. The paper also provides a representation formula for the effective Hamiltonian and gives illustrations to calculus of variations, averaging and one-dimensional problems
Semilinear Elliptic Equations on Thin Network-Shaped Domains with Variable Thickness

AbstractWe study semilinear elliptic equations in a thin domain which is shaped like a network and degenerate into a geometric graph when a certain parameter tends to zero. It is shown by using a comparison technique that a solution of the equation in the network-shaped domain approaches a solution of an associated limit equation on the graph. Conversely, when the limit equation on the graph has a solution, we prove the existence of a solution of the PDE in the network-shaped domain which converges to the solution of the limit equation
Maslov index, Poincaré–Birkhoff Theorem and Periodic Solutions of Asymptotically Linear Planar Hamiltonian Systems

AbstractWe study the relationship between the twist condition in the Poincaré–Birkhoff fixed point theorem and the assumptions on the Maslov index for asymptotically linear planar Hamiltonian systems. For this aim, we develop a variant of the Poincaré–Birkhoff theorem which, together with its classical version, allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the Maslov indexes associated to the linearizations of the planar system at zero and at infinity
Pointwise Green's Function Bounds for Multidimensional Scalar Viscous Shock Fronts

AbstractWe determine detailed pointwise bounds for the Green's function of the linearized operator about a multidimensional scalar viscous shock front. These extend the pointwise semigroup methods introduced by Howard and Zumbrun in the one-dimensional case to multidimensions, sharpening Lp estimates obtained by Goodman and Miller using a weighted norm approach. Moreover, our results apply to shocks of arbitrary strength, as previous results did not. As described in a companion paper, the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear Lp-asymptotic behavior of the front under small perturbation. The analysis of the multidimensional case involves several new features not found in the one-dimensional case, concerned with the geometry of propagating signals
Integrability and Linearizability of the Lotka–Volterra System with a Saddle Point with Rational Hyperbolicity Ratio

AbstractIn this paper, we consider normalizability, integrability and linearizability properties of the Lotka–Volterra system in the neighborhood of a singular point with eigenvalues 1 and −λ. The results are obtained by generalizing and expanding two methods already known: the power expansion of the first integral or of the linearizing transformation and the transformation of the saddle into a node. With these methods we find conditions that are valid for λ∈R+ or λ∈Q. These conditions will allow us to find all the integrable and linearizable systems for λ=p2 and 2p with p∈N+
Initial–Boundary Value Problem for a Class of Linear Relaxation Systems in Arbitrary Space Dimensions

AbstractIn this paper, we consider the characteristic initial–boundary value problem (IBVP) for the multi-dimensional Jin–Xin relaxation model in a half-space with arbitrary space dimension n⩾2. As in the one-dimensional case (n=1, see (J. Differential Equations, 167 (2000), 388–437), our main interest is on the precise structural stability conditions on the relaxation system, particularly the formulation of boundary conditions, such that the relaxation IBVP is stiffly well posed, that is, uniformly well posed independent of the relaxation parameter ε>0, and the solution of the relaxation IBVP converges, as ε→0, to that of the corresponding limiting equilibrium system, except for a sharp transition layer near the boundary. Our main result can be roughly stated as Stiff Kreiss Condition=Uniform Kreiss Condition for the relaxation IBVP we consider in this paper, which is in sharp contrast to the one-dimensional case (Z. Xin and W.-Q. Xu, J. Differential Equations, 167 (2000), 388–437). More precisely, we show that the Uniform Kreiss Condition (which is necessary and sufficient for the well posedness of the relaxation IBVP for each fixed ε), together with the subcharacteristic condition (which is necessary and sufficient for the stiff well posedness of the corresponding Cauchy problem), also guarantees the stiff well posedness of our relaxation IBVP and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate. Optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified
Homoclinic Solutions for Swift–Hohenberg and Suspension Bridge Type Equations

AbstractWe establish the existence of homoclinic solutions for a class of fourth-order equations which includes the Swift–Hohenberg model and the suspension bridge equation. In the first case, the nonlinearity has three zeros, corresponding to a double-well potential, while in the second case the nonlinearity is asymptotically constant on one side. The Swift–Hohenberg model is a higher-order extension of the classical Fisher–Kolmogorov model. Its more complicated dynamics give rise to further possibilities of pattern formation. The suspension bridge equation was studied by Chen and McKenna (J. Differential Equations136 (1997), 325–355); we give a positive answer to an open question raised by the authors
Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations

AbstractUsing the maximum principle for semicontinuous functions (Differential Integral Equations3 (1990), 1001–1014; Bull. Amer. Math. Soc. (N.S)27 (1992), 1–67), we establish a general “continuous dependence on the non- linearities” estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by Souganidis (J. Differential Equations56 (1985), 345–390) for first- order Hamilton–Jacobi equations and a recent result by Cockburn et al. (J. Differential Equations170 (2001), 180–187) for a class of degenerate parabolic second–order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) L∞ and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)–(iii) on the Hamilton–Jacobi– Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here
The Semiclassical Limit of the Nonlinear Schrödinger Equation in a Radial Potential

AbstractIn this paper, we are concerned with the following nonlinear Schrödinger equation: ih̵∂ψ∂t=−h̵22mΔψ+V(x)ψ−γh̵∣ψ∣p−2ψ, γh̵>0, x∈R2, where ħ>0, 2<p<6, ψ : R2→C, and the potential V is radially symmetric. Our main purpose is to obtain positive solutions among the functions having the form ψ(r,θ,t)=exp(Imh̵θ/h̵+iEt/h̵)v(r), being r,θ the polar coordinates in the plane. Since we assume Mħ>0, the functions in this special class have nontrivial angular momentum as it will be specified in the Introduction. Furthermore, our solutions exhibit a “spike-layer” pattern when the parameter ħ approaches zero; our object is to analyse the appearance of such type of concentration asymptotic behaviour in order to locate the asymptotic peaks