Showing papers in "Discrete and Continuous Dynamical Systems in 1998"

On the global existence of a cross-diffusion system

Abstract: We consider a strongly-coupled nonlinear parabolic system which arises from population dynamics. The global existence of classical solutions is established when the space dimension is two and one of the cross-diffusion pressures is zero.

Regularity of the composition operator in spaces of Hölder functions

Abstract: We study the regularity of the composition operator $((f, g)\to g \circ f)$ in spaces of Holder differentiable functions. Depending on the smooth norms used to topologize $f, g$ and their composition, the operator has different differentiability properties. We give complete and sharp results for the classical Holder spaces of functions defined on geometrically well behaved open sets in Banach spaces. We also provide examples that show that the regularity conclusions are sharp and also that if the geometric conditions fail, even in finite dimensions, many elements of the theory of functions (smoothing, interpolation, extensions) can have somewhat unexpected properties.

Dynamics of a piecewise rotation

Abstract: We investigate the dynamics of systems generalizing interval exchanges to planar mappings. Unlike interval exchanges and translations, our mappings, despite the lack of hyperbolicity, exhibit many features of attractors. The main result states that for a certain class of noninvertible piecewise isometries, orbits visiting both atoms infinitely often must accumulates on the boundaries of the attractor consisting of two maximal invariant discs $D_0 \cup D_1$ fixed by $T$. The key new idea is a dynamical and geometric observation about the monotonic behavior of orbits of a certain first-return map. Our model emerges as the local map for other piecewise isometries and can be the basis for the construction of more complicated molecular attractors.

Sensitivity analysis for state constrained optimal control problems

Abstract: A family of parameter dependent optimal control poblems for nonlinear ODEs is considered. The problems are subject to pointwise control and state inequality type constraints. It is assumed that, at the reference value of the parameter the reference optimal solution exists and is regular. Regularity conditions are formulated under which the original problems are locally equivalent to some problems subject to equality type constraints only. The classical implicit function theorem is applied to these new problems to investigate Frechet differentiability of the solutions with respect to the parameter. A numerical example is provided.

Monotonicity and convergence results in order-preserving systems in the presence of symmetry

Abstract: This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.

Positive perturbation of operator semigroups: Growth bounds, essential compactness, and asynchronous exponential growth

Abstract: If $B$ is the generator of an increasing locally Lipschitz continuous integrated semigroup on an abstract L space $X$ and $C: D(B) \to X$ perturbs $B$ positively, then $A = B + C$ is again the generator of an increasing l.L.c. integrated semigroup. In this paper we study the growth bound and the compactness properties of the $C_0$ semigroup $S_\circ$ that is generated by the part of $A$ in $X_\circ = \overline {D(B)}$. We derive conditions in terms of the resolvent outputs $ F(\lambda) = C (\lambda - B)^{-1} $ for the semigroup $S_\circ$ to be eventually compact or essentially compact and to exhibit asynchronous exponential growth. We apply our results to age-structured population models with additional structures. We consider an age-structured model with spatial diffusion and an age-size-structured model.

Minimal sets of periods for torus maps

Abstract: Let $T^r$ be the $r$-dimensional torus, and let $f:T^r\to T^r$ be a map. If $\Per(f)$ denotes the set of periods of $f$, the minimal set of periods of $f$, denoted by $\MPer(f)$, is defined as $\bigcap_{g\cong f}\Per(g)$ where $g:T^r\to T^r$ is homotopic to $f$. First, we characterize the set $\MPer(f)$ in terms of the Nielsen numbers of the iterates of $f$. Second, we distinguish three types of the set $\MPer(f)$ and show that for each type and any given dimension $r$, the variation of $\MPer(f)$ is uniformly bounded in a suitable sense. Finally, we classify all the sets $\MPer(f)$ for self-maps of the $3$-dimensional torus.

On the initial boundary value problem for the damped Boussinesq equation

Abstract: The first initial-boundary value problem for the damped Boussinesq equation $ u_{t t}-2bu_{t x x}=-\alpha u_{x x x x}+u_{x x}+\beta (u^2)_{x x}, x\in (0,\pi ),\quad t>0,$ with $\alpha, b=const>0,\quad \beta =const\in R^1,$ is considered with small initial data. For the most interesting case $\alpha >b^2$ corresponding to an infinite number of damped oscillations its solution is constructed in the form of a Fourier series which coefficients in their own turn are represented as series in small parameter present in the initial conditions. The solution of the corresponding problem for the classical Boussinesq equation on $[0,T],\quad T<+\infty,$ is obtained by means of passing to the limit $b\rightarrow +0.$ Long-time asymptotics of the solution in question is calculated which shows the presence of the damped oscillations decaying exponentially in time. This is in contrast with the long time behavior of the solution of the periodic problem studied in [30] which major term increases linearly with time.

Semidiscretization in time for nonlinear Schrödinger-waves equations

Abstract: In this paper, we are concerned with Crank-Nicolson like schemes for: $ (NLW_\omega ) \frac{1}{\omega^2} \partial_t^2 E_\omega -i\partial_t E_\omega -\D E_\omega =\lambda | E_\omega |^{2\sigma} E_\omega. $ We present two schemes for which we give some convergence results. On of the scheme is dissipative and we describe precisely the dissipation. We prove that the solution of the second scheme fits that of $(NLW_\omega )$ while the first one compute a average value of the solution.

Optimal energy decay rate in a damped Rayleigh beam

Abstract: We consider a clamped Rayleigh beam subject to a positive viscous damping. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the underlying system. We next identify the optimal energy decay rate of the system with the supremum of the real part of the spectrum of the infinitesimal generator of the associated semigroup.

Recession methods for equilibrium problems and applications to variational and hemivariational inequalities

Abstract: In this paper, we give some existence results for equilibrium problems by proceeding to a perturbation of the initial problem and using techniques of recession analysis. We develop and describe thoroughly recession condition which ensure existence of at least one solution for hemivariational inequalities introduced by Panagiotopoulos. Then we give two applications to resolution of concrete variational inequalities. We shall examine two examples. The first one concerns the unilateral boundary condition. In the second, we shall consider the contact problem with given friction on part of the boundary.

Numerical approximation of a parabolic problem with a nonlinear boundary condition

Abstract: We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an approximation of the original problem. Our objective is to analyze whether this system has a similar behaviour than the original problem. We find a necessary and sufficient condition for blow up of this system. However, this condition is slightly different than the one known for the original problem, in particular, there are cases in which the continuous problem has blow up while its semidiscrete approximation does not. Under certain assumptions we also prove that the numerical blow up time converges to the real blow-up time when the meshsize goes to zero. Our proofs are given in one space dimension. Similar arguments could be applied for higher dimensions but a further analysis is required.

On two-dimensional Riemann problem for pressure-gradient equations of the Euler system

Abstract: We consider the two-dimensional Riemann problem for the pressure-gradient equations with four pieces of initial data, so restricted that only one elementary wave appears at each interface. This model comes from the flux-splitting of the compressible Euler system. Lack of the velocity in the eigenvalues, the slip lines have little influence on the structures of solutions. The flow exhibits the simpler patterns than in the Euler system, which makes it possible to clarify the interaction of waves in two dimensions. The present paper is devoted to analyzing the structures of solutions and presenting numerical results to the two-dimensional Riemann problem. Especially, we give the criterion of transition from the regular reflection to the Mach reflection in the interaction of shocks.

Anti-periodic solutions to a class of non-monotone evolution equations

Abstract: We establish the existence of solutions to an anti-periodic non-monotone boundary value problem. Our approach relies on a combination of monotonicity and compactness methods.

Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two

Abstract: Nonlinear stability and some other dynamical properties for a KS type equation in space dimension two are studied in this article. We consider here a variation of the KS equation where the derivatives in the nonlinear and the antidissipative linear terms are in one single direction. We prove the nonlinear stability for all positive times and study the corresponding attractor.

Approximation error for invariant density calculations

Abstract: Let $T:[0,1]\rightarrow[0,1]$ be an expanding piecewise-onto map with bounded distortion and countably many intervals of monotonicity. We prove a uniform bound on the rate of exponential convergence to equilibrium for iterates of the Perron-Frobenius operator. The quantitative information thus obtained is applied to prove explicit error bounds for Ulam's method for approximating invariant measures. The approach also yields rates of mixing for the matrix representations of Ulam approximation. "Monte-Carlo" type simulations of the scheme are discussed. The method of proof is applicable to multi-dimensional transformations, although the only generalisations presented here are to a limited class of "non-onto" one-dimensional transformations.

Exponential attractors for nonautonomous first-order evolution equations

Abstract: Our aim in this article is to study the existence of exponential attractors for nonautonomous dissipative evolution equations. We follow the approach of Chepyzhov and Vishik, which consists in studying a semigroup on an extended space.

Topological mapping properties defined by digraphs

Abstract: Topological transitivity, weak mixing and non-wandering are definitions used in topological dynamics to describe the ways in which open sets feed into each other under iteration. Using finite directed graphs, these definitions are generalized to obtain topological mapping properties. The extent to which these mapping properties are logically distinct is examined. There are three distinct properties which entail "interesting" dynamics. Two of these, transitivity and weak mixing, are already well known. The third does notappear in the literature but turns out to be close to weak mixing in a sense to be discussed. The remaining properties comprise a countably infinite collection of distinct properties entailing somewhat less interesting dynamics and including non-wandering.

Applied symbolic dynamics: attractors and filtrations

Abstract: This paper is a study of the global structure of the attractors of a dynamical system. The dynamical system is associated with an oriented graph called a Symbolic Image of the system. The symbolic image can be considered as a finite discrete approximation of the dynamical system flow. Investigation of the symbolic image provides an opportunity to localize the attractors of the system and to estimate their domains of attraction. A special sequence of symbolic images is considered in order to obtain precise knowledge about the global structure of the attractors and to get filtrations of the system.

The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order

Abstract: We show the local in time solvability of the Cauchy problem for nonlinear wave equations in the Sobolev space of critical order with nonlinear term of exponential type.

Homoclinics and complex dynamics in slowly oscillating systems

Abstract: This paper deals with a class of second order dynamical systems with slowly oscillating coefficients, see $(1)$. Using variational methods, perturbative in nature, we show that $(1)$ has multi-bump homoclinics and a complex dynamics.

On the behaviour of the solutions for a class of nonlinear elliptic problems in exterior domains

Abstract: We state a result concerning the limit of a class of minimization problems. This result is applied to describe the asymptotic behaviour of the solutions of an elliptic Dirichlet problem in exterior domains $\Omega$ of $\mathbb{R}^N$, when $\mathbb{R}^N$ \ $Omega$ becomes larger and larger.

Multiple solutions results for two-point boundary value problem with resonance

Abstract: In this paper we prove the existence of multiple solutions for two-point boundary value problem, which is resonant type both at origin and at infinity, and with higher eigenvalue and unbounded nonlinear terms. Our main ingredients are Morse theory and the computations of critical groups.

Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation

Abstract: We study the Cauchy problem for a nonlinear Schrodinger equation which is the generalization of a one arising in plasma physics. We focus on the so called subcritical case and prove that when the initial datum is "small", the solution exists globally in time and decays in time just like in the linear case. For a certain range of the exponent in the nonlinear term, we prove that the solution is asymptotic to a "final state" and the nonexistence of asymptotically free solutions. The method used in this paper is based on some gauge transformation and on a certain phase function.

Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation

Abstract: In this paper, we first prove existence results for general systems of differential equations of parabolic and hyperbolic type in a Hilbert space setting using the notion of Agmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessary and sufficient condition on the boundary and/or transmission conditions which insures the dissipativity of the (spatial) operators. Our second goal is to take advantage of the one-dimensional structure of networks in order to build appropriate prewavelet bases in view to the numerical approximation of the above problems. Indeed we show that the use of such bases for their approximation (by the Galerkin method for elliptic operators and a fully discrete scheme for parabolic ones) leads to linear systems which can be preconditioned by a diagonal matrix and then can be reduced to systems with a condition number uniformly bounded (with respect to the mesh parameter).

Longtime behavior of a homogenized model in viscoelastodynamics

Abstract: A material with heterogeneous structure at microscopic level is considered. The microscopic mechanical behavior is described by a stress-strain law of Kelvin-Voigt type. It has been shown that a homogenization process leads to a macroscopic stress-strain relation containing a time convolution term which accounts for memory effects. Consequently, the displacement field $\mathbf{u}$ obeys to a Volterra integrodifferential motion equation. The longtime behavior of $\mathbf{u}$ is here investigated proving the existence of a uniform attractor when the body forces vary in a suitable metric space.

Remarks on determining projections for stochastic dissipative equations

Abstract: In this paper we consider the notion of determining projections for two classes of stochastic dissipative equations: a reaction-diffusion equation and a 2-dimensional Navier-Stokes equation. We define certain finite dimensional objects that can capture the asymptotic behavior of the related dynamical system. They are projections on a space of polynomial functions, generalizing the classical (but not very much studied in a stochastic context) concepts of determining modes, nodes and volumes.

Periodic systems for the higher-dimensional Laplace transformation

Abstract: We consider overdetermined systems of linear partial differential equations of the form $ y_{,k\l}+a_{k\l}^ky_{,k}+a_{k\l}^\ly_{,\l}+c_{k\l}y=0\ , \quad 1\le k e\l\le n\ , $ where the coefficients are smooth functions satisfying certain integrability conditions. Generalizing the classical theory of second order linear hyperbolic partial differential equation in the plane, we consider higher-dimensional Laplace invariants of a system of the above class. These invariants are characterized as functions which must satisfy a set of differential equations. We establish a normal form for any system of the above class in terms of these invariants. Moreover, we solve the periodicity problem for the higher-dimensional Laplace transformation applied to such systems, generalizing a classical theorem of Darboux which shows that for $n=2$ a 1-periodic equation is equivalent to the Klein-Gordon equation.

Subharmonic solutions for second order Hamiltonian systems

Abstract: In this paper, we are interested in the existence of subharmonic solutions for the problem $ u_{t t} + G'(u) = f(t), $ where $G:R^{ N} \rightarrow R$ is not necessarily convex and $f:R \rightarrow R^N$ is periodic with minimal period $T > 0$.

Invariants of twist-wise flow equivalence

Abstract: Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.