Traveling Monotonic Fronts in the Discrete Nagumo Equation

We give an alternative proof of the theorem, which states that no propagation failure occurs for the discrete Nagumo equation with “translationally invariant” stationary monotonic fronts. The theorem was recently proved with the use of the invariant manifolds for lattice differential equations by Hupkes, Pelinovsky, and Sandstede. The alternative proof relies on the analysis of the advance-delay operator associated with the translationally invariant stationary front. This operator exhibits an infinite-dimensional kernel spanned by Fourier harmonics of front’s translations, which are accounted when the stationary front is continued into the traveling one.     

Geometric Proof of Lie's Linearization Theorem

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.     

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

We study the existence and nonexistence of traveling waves of a general diffusive Kermack–McKendrick SIR model with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform. Our study provides a promising method to deal with high dimensional epidemic models.     

Floating Bodies in Neutral Equilibrium

In his paper preceding in this issue, Finn proved that if the contact angle γ of a convex body B{\mathcal{B}} with a given liquid is π/2, and if B{\mathcal{B}} can be made to float in “neutral equilibrium” in the liquid in any orientation, then B{\mathcal{B}} is a metric ball. The present work extends that result, with an independent proof, to any contact angle in the range 0 < γ < π. Our result is equivalent to the general geometric theorem that if for every orientation of a plane, it can be translated to meet a given strictly convex body B{\mathcal{B}} in a fixed angle γ within the above range, then B{\mathcal{B}} is a metric ball.     

Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L^p for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.     

A Simplified Proof of a Liouville Theorem for Nonnegative Solution of a Subcritical Semilinear Heat Equations

We give a new proof of the Liouville theorem proved by Merle and Zaag for nonnegative solutions of the semilinear heat equation with power nonlinearity. Our proof has a pedagogical interest and is based on Kaplan’s blow-up criterion.      

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS (Ⅱ)─PART 2

In the part 2, theorem 3.1 studied in part 1^[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differential equations to a form concerning standard systems of equations in the theory of differentiable dynamical systems. Then by using theorem 3.1 together with the preliminary theorem 2.1, the main theorem of this paper announced in part 1 is proved. The definition of admissible perturbation is contained in the appendix of part 2. The meanings of the main theorem is described in the introduction of part 1.     

A Correction of an Inconsistency in my Paper “Cauchy's Theorem on Manifolds”

The Cauchy postulates are required for the formulation and proof of Cauchy's theorem for the existence of stress. The generalized postulates and theorem in the geometric setting of differentiable manifolds was considered in a previous paper. This note presents an inconsistency in one of the proposed postulates, the boundedness postulate, and corrects it by specifying a weaker requirement.     

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.     

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.     

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS (Ⅱ)──PART 2

IV.ChangingVariablesTheproofofthemaintheoremstatedinSectionIofthepaperwillbecompletedinSectionVI.Now,withthesystem.)giveninSectionill,letuswritesimplyTheexistenceofsuchf(k)followsfromtheproperty(II)ofthesystem.).Theexistenceisunique.Thenbytheproperty(II)o…     

A Topological Proof of Stability of N-Front Solutions of the FitzHugh–Nagumo Equations

Consideration is devoted to traveling N-front wave solutions of the FitzHugh–Nagumo equations of the bistable type. Especially, stability of the N-front wave is proven. In the proof, the eigenvalue problem for the N-front wave bifurcating from coexisting simple front and back waves is regarded as a bifurcation problem for projectivised eigenvalue equations, and a topological index is employed to detect eigenvalues.     

SET-VALUED CARISTI’S FIXED POINT THEOREM AND EKELANDS VARIATIONAL PRINCIPLE

This paper proposes a formally stronger set-valued Caristi’s fixed point theorem and by using a simple method we give a direct proof for the equivalence between Ekeland’s variational principle and thin set-valued Caristi’s fixed point theorem. The results stated in this paper improve and strengthen the corresponding results in [4].     

Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.     

On the dynamics of an elastic body containing a moving crack

It is proved that the energy release rate and the rate of entropy production in the dynamics of an elastic body containing a moving crack are proportional. Moreover, a theorem of the domain of influence type and a uniqueness theorem for solutions to the boundary-initial-value problem of brittle fracture mechanics are proved.     

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov–Schmidt reduction through Lin’s method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.     

Universality of Crystallographic Pinning

We study traveling waves for reaction diffusion equations on the spatially discrete domain \mathbb Z²{\mathbb Z^2}. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. In [19] it was shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth, which itself was considered in [6]. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.     

A comment on the proof of Fermat's last theorem

In this paper, some conmments on the proof of Fermat’s last theorem are proposed.The main resuilt is thai the proof proposed by Wong Chiahe is only part of proof for fermat’s last theorem. That is to sqy,the proof is not all-full proof to Fermat’s last theorem.     

The axisymmetric problem with initial conditions for the equations of motion of an ideal incompressible fluid

The present paper presents a proof of the existence and uniqueness theorem for the solution of the axisymmetric problem with initial conditions for the Euler equations in the case of an incompressible fluid. We consider the case of the nonporous wall, and also the transpiration problem in the formulation given in [1]. Global unique solvability is proved for assumptions only on the smoothness of the conditions and for all values of the time t. The existence theorem for a small time segment in the case of a nonporous wall has been proved for the general three-dimensional problem in [2, 3]. For the proof we use a method analogous to that developed in [1] for planar flows. The a priori estimate of the vorticity which is used in the present study was obtained previously in [4],The author wishes to thank V. I. Yudovich for continued interest in the study and many valuable suggestions.