Resonant almost periodic oscillations in systems with slow varying parameters

We consider two-dimensional systems with fast rotating phase and slow varying parameters and assume that the right-hand side of a system almost periodically depends on fast and slow times. We investigate the conditions of the existence and stability of resonant almost periodic solutions. As an example, we consider forced oscillations of a mathematical pendulum under the action of a sum of two small forces with closed frequencies.     

Stability of the stationary solutions of the Allen–Cahn equation with non-constant stiffness

We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We assume the stiffness to be a positive function of the field and we discuss the stability of the stationary solutions proving both linear and local non-linear stability.     

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.     

Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation

We study the global-in-time behavior of solutions to a reaction–diffusion system with mass conservation, as proposed in the study of cell polarity, particularly, the second model of the work by Otsuji et al. (PLoS Comput Biol 3:e108, 2007). First, we show the existence of a Lyapunov function and confirm the global-in-time existence of the solution with compact orbit. Then we study the stability and instability of stationary solutions by using the semi-unfolding-minimality property and the spectral comparison. As a result the dynamics near the stationary solutions is qualitatively characterized by a variational function.     

Moore-Gibson-Thompson theory for thermoelastic dielectrics

We consider the system of equations determining the linear thermoelastic deformations of dielectrics within the recently called Moore-Gibson-Thompson(MGT)theory.First,we obtain the system of equations for such a case.Second,we consider the case of a rigid solid and show the existence and the exponential decay of solutions.Third,we consider the thermoelastic case and obtain the existence and the stability of the solutions.Exponential decay of solutions in the one-dimensional case is also recalled.     

A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations

In this paper, we study the structural stability of the Cahn-Hilliard equation and the phase-field equations. We show that the Cahn-Hilliard equation and the phase-field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold; particularly, the flow nearby hyperbolic stationary solutions is structurally stable.     

Bifurcation and stability of spatially periodic solutions of nonlinear evolution equations with integral operators

A more general kind of nonlinear evolution equations with integral operators is discussed in order to study the spatially periodic static bifurcating solutions and their stability. At first, the necessary condition and the sufficient condition for the existence of bifurcation are studied respectively. The stability of the equilibrium solutions is analyzed by the method of semigroups of linear operators. We also obtain the principle of exchange of stability in this case. As an example of application, a concrete result for a special case with integral operators of exponential type is presented. This work was supported by the Chinese National Foundation of Natural Science     

Traveling Waves for a Biological Reaction-Diffusion Model

We investigate the existence of traveling wave solutions for a system of reaction–diffusion equations that has been used as a model for microbial growth in a flow reactor and for the diffusive epidemic population. The existence of traveling waves was conjectured early but only has been proved recently for sufficiently small diffusion coefficient by the singular perturbation technique. In this paper we show the existence of traveling waves for an arbitrary diffusion coefficient. Our approach is a shooting method with the aid of an appropriately constructed Liapunov function.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.Wenzhang Huang-Research was supported in part by NSF Grant DMS-0204676.     

On steady distributions of self-attracting clusters under friction and fluctuations

We consider the Vlasov-Fokker-Planek equation with a Newtonian, attracting potential and study its stationary solutions, given by the generalized Lane-Emden equation. In a two-dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse. For a one-dimensional model we prove some results on existence, uniqueness, stability and symmetry-breaking of stationary solutions.     

Regularity and stability for a viscoelastic material with a singular memory kernel

We consider a linear viscoelastic material whose relaxation function may exhibit an initial singularity. We show that the Laplace transform method is still applicable in order to study existence, uniqueness and asymptotic behaviour of the solution to the dynamic problem. In order to provide these results, we impose on the relaxation function only restrictions deriving from Thermodynamics. Moreover, by using energy estimates, we establish a stability theorem. Finally, for a class of singular kernels, we obtain a regularity result which ensures the asymptotic stability of the solution.This work is supported by G.N.F.M. of C.N.R. and by M.U.R.S.T. 40% and 60% projects.     

Second Order Abstract Neutral Functional Differential Equations

In this paper we are concerned with a class of second order abstract neutral functional differential equations with finite delay in a Banach space. We establish the existence of mild and classical solutions for the nonlinear equation, and we show that the map defined by the mild solutions of the linear equation is a strongly continuous semigroup of bounded linear operators on an appropriate space. We use this semigroup to establish a variation of constants formula to solve the inhomogeneous linear equation.     

Nonlinear Elliptic–Parabolic Problems

We introduce a notion of viscosity solutions for a general class of elliptic–parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new, even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in Alt and Luckhaus (Math Z 183:311–341, 1983).     

Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems

In this paper, we study the existence, uniqueness and stability of the periodic solutions for fourth-order nonlinear nonhomogeneous periodic systems with slowly changing coefficients by using the method of Liapunor Function. We obtain some sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions of these systems and estimate the extent to which the coefficients are allowed to change.     

The Principal Floquet Bundle and Exponential Separation for Linear Parabolic Equations

We consider linear nonautonomous second order parabolic equations on bounded domains subject to Dirichlet boundary condition. Under mild regularity assumptions on the coefficients and the domain, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. Our main theorem extends in a natural way standard results on principal eigenvalues and eigenfunctions of elliptic and time-periodic parabolic equations. Similar theorems were earlier available only for smooth domains and coefficients. As a corollary of our main result, we obtain the uniqueness of positive entire solutions of the equations in     

A class of Kolmogorov's ecological system with prey having constant adding rate

In this paper, by using the qualitative method, we study a class of Kolmogorov's ecological system with prey having constant adding rate, discuss the relative position and the character of the equilibriums, the global stability of the practical equilibriums and give a group of conditions for the boundedness of the solutions, the nonexistence, the existence and the uniqueness of the limit cycle of the system. Most results obtained in papers [1] and [2] are included or generalized.     

Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.     

Existence of Traveling-Wave Solutions for Hyperbolic Systems of Balance Laws

This paper is concerned with traveling-wave solutions for hyperbolic systems of balance laws satisfying a stability condition and a Kawashima-like condition. We are interested in the case where the traveling-wave equations have a singularity, which is absent for 2 × 2 systems satisfying the two conditions. To deal with the singularity, we reduce the problem to a parametrized one without singularity by using the center manifold theorem. For the parametrized problem, we prove the existence of solutions by modifying an existing argument in the literature. In this way, we show the existence of traveling-wave solutions.     

Stability of solutions of one class of nonlinear dynamic equations

We study the stability of the zero solution of a nonlinear dynamic equation on a time scale under certain assumptions on the right-hand side of this equation. In addition to conditions for the existence and uniqueness of a solution of the Cauchy problem, we also assume that the exponential function of the linear approximation is bounded, and the norms of the nonlinear part and its derivatives with respect to the components of the space variable are majorized by power functions of the norm of the space variable. Using the generalized method of Lyapunov functions, we obtain sufficient conditions for the stability of the zero solution of the nonlinear equation under consideration.     

Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with <Emphasis Type="Italic">n</Emphasis>-Degrees of Freedom in the Case of Single Resonance

This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.