Peter Takáč

Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups

Abstract Convergence to equilibrium for every positive semi-orbit of strongly increasing discrete-time semigroups on strongly ordered spaces is proved. The domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered Banach space which is continuously imbedded into a Banach lattice. The semigroup is assumed to be order-compact, and every positive semi-orbit is assumed to be order-bounded. The crucial hypothesis is the Ljapunov stability of all equilibria. It is also proved that the set of equilibria is a simply ordered arc. The key tools are invariant d -hypersurfaces.

Dispersion population models discrete in time and continuous in space

We analyze a discrete-time model of populations that grow and disperse in separate phases. The growth phase is a nonlinear process that allows for the effects of local crowding. The dispersion phase is a linear process that distributes the population throughout its spatial habitat. Our study quantifies the issues of survival and extinction, the existence and stability of nontrivial steady states, and the comparison of various dispersion strategies. Our results show that all of these issues are tied to the global nature of various model parameters. The extreme strategies of staying-in place and going-everywhere-uniformly are compared numerically to diffusion strategies in various contexts. We approach the mathematical analysis of our model from a functional analysis and an operator theory point of view. We use recent results from the theory of positive operators in Banach lattices.

Dynamics of the Ginzburg-Landau equations of superconductivity

This article is concerned with the dynamical properties of solutions of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. It is shown that the TDGL equations define a dynamical process when the applied magnetic field varies with time, and a dynamical system when the applied magnetic field is stationary. The dynamical system describes the large-time asymptotic behavior: Every solution of the TDGL equations is attracted to a set of stationary solutions, which are divergence free. These results are obtained in the {open_quotes}{phi} = -{omega}({gradient}{center_dot}A){close_quotes} gauge, which reduces to the standard {close_quotes}{phi} = -{gradient}{center_dot}A{close_quotes} gauge if {omega} = 1 and to the zero-electric potential gauge if {omega} = 0; the treatment captures both in a unified framework. This gauge forces the London gauge, {gradient}{center_dot}A = 0, for any stationary solution of the TDGL equations.

A counterexample to the Fredholm alternative for the -Laplacian

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional p-Laplacian with 1 < p < ∞ is considered: −(|u′|p−2u′)′ − λ|u|p−2u = f(x) for 0 < x < T ; u(0) = u(T ) = 0, (*) where f ≡ 1 + h with h ∈ L∞(0, T ) small enough. Solvability properties of Problem (*) with respect to the spectral parameter λ ∈ R are investigated. We focus our attention on some fundamental differences between the cases p 6= 2 and p = 2. For p 6= 2 we give a counterexample to the classical Fredholm alternative (which is valid for the linear case p = 2).

Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry

Given a strongly monotone discrete-time dynamical system {Tn: X → X: n ϵ Z+} in an open and order-convex subset X of a separable strongly ordered Banach space V and a compact connected metrizable group G whose action on X is monotone and commutes with T, we prove under some reasonable additional hypotheses that the ω-limit set ω(x) of every stable point x ϵ X consists of symmetric points only, i.e., g · w = w for all w ϵ ω(x) and g ϵ G. Moreover, the set of all unstable points is contained in a union of at most countably many Lipschitz manifolds of codimension one in V where each manifold is invariant under T and the action of G. This result is applied to the time-periodic, spatially independent, irreducible cooperative system of n reaction-diffusion equations ∂u∂t = D(t) Δu + F(t, u) for (t, x) ϵ R1+ × RN with spatially periodic boundary conditions in RN, and with an initial distribution u0 which is continuous and satisfies the boundary conditions. If u0 is stable, then ω(u0) contains spatially constant functions only, and the dynamics on ω(u0) is given by the irreducible cooperative system of n ODEs dudt = F(t, u) for t ϵ R1+. Here T is the Poincare map. If n = 1 then ω(u0) is a single fixed point of T; if n = 2 then ω(v0) is a single fixed point of T for each v0 ϵ ω(u0).

A construction of stable subharmonic orbits in monotone time-periodic dynamical systems

AbstractWe consider a family of abstract semilinear elliptic-like equationsB(t,uo(t))=0 for an unknown functionu0(t) parametrized by the time-variablet≥0 and valued in a Banach spaceX. Suppose that bothB(.,u) andu0 areT-periodic in timet, and each Fréchet derivative

FRONT PROPAGATION IN NONLINEAR PARABOLIC EQUATIONS

- A_B (t) \equiv \frac{{\partial B}}{{\partial u}}(t,u_0 (t)),t \geqslant 0

On maximum and comparison principles for parabolic problems with the p-Laplacian

generates an exponentially decaying, analyticC0-semigroup inX. We show that, for every small ε>0, the abstract parabolic-like evolution equationεduε/dt=B(t,uε(t)),t≥0, has a linearly stableT-periodic solutionuε nearu0. Given any integern≥2, we construct examples ofB andu0 such that the minimum periods ofB(.,u) andu0, respectively, areτ=T/n andT. Thenuε(t), t≥0, is alinearly stable subharmonic orbit of minimum periodT for our τ-periodic evolution equation. The corresponding dynamical systems are strongly monotone.