We consider a nonlinear Schrodinger equation (NLS) posed on a graph (or network) composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of d-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass (L-2-norm). When existing, the minimizer is called ground state and it is the profile of an orbitally stable standing wave for the NLS evolution. We prove that a ground state exists for sufficiently small masses whenever the quadratic part of the energy admits a simple isolated eigenvalue at the bottom of the spectrum (the linear ground state). This is a wide generalization of a result previously obtained for a star-graph with a single vertex. The main part of the proof is devoted to prove the concentration compactness principle for starlike structures; this is non trivial due to the lack of translation invariance of the domain. Then we show that a minimizing, bounded, H-1 sequence for the constrained NLS energy with external linear potentials is in fact convergent if its mass is small enough. Moreover we show that the ground state bifurcates from the vanishing solution at the bottom of the linear spectrum. Examples are provided with a discussion of the hypotheses on the linear part.

In this paper we deal with nonlinear differential systems of the form x'(t) = Sigma(k)(i=0) epsilon(i) F(i()t,x) + epsilon(k+1) R(t,x,epsilon), where F-i : R x D -> R-n for i = 0, 1, ... , k, and R : R x D x (-epsilon 0, epsilon 0). R-n are continuous functions, and T - periodic in the first variable, D being an open subset of R-n, and epsilon a small parameter. For such differential systems, which do not need to be of class C-1, under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in epsilon. Some applications are also performed.

A continuous map f from a compact interval I into itself is densely (resp. generically) chaotic if the set of points (x, y) such that lim supn ->+infinity f(n)(x) - f(n)(y) - > 0 and lim inf(n ->+infinity) vertical bar f(n)(x) - fn(y)vertical bar = 0 is dense (resp. residual) in I x I. We prove that if the interval map f is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which contains a horseshoe for f(2). It implies that every densely chaotic interval map is of type at most 6 for Sharkovskii's order (i.e. there exists a periodic point of period 6), and its topological entropy is at least (log 2)/2. We show that equalities can be obtained.

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtain improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present a few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the convergence analysis of the data assimilation algorithm.

To understand the spreading and interaction of two-competing species, we study the dynamics for a two-species competition-diffusion model with two free boundaries. Here, the two free boundaries which describe the spreading fronts of two competing species, respectively, may intersect each other. Our result shows there exists a critical value such that the superior competitor always spreads successfully if its territory size is above this constant at some time. Otherwise, the superior competitor can be wiped out by the inferior competitor. Moreover, if the inferior competitor does not spread fast enough such that the superior competitor can catch up with it, the inferior competitor will be wiped out eventually and then a spreading-vanishing trichotomy is established. We also provide some characterization of the spreading-vanishing trichotomy via some parameters of the model. On the other hand, when the superior competitor spreads successfully but with a sufficiently low speed, the inferior competitor can also spread successfully even the superior species is much stronger than the weaker one. It means that the inferior competitor can survive if the superior species cannot catch up with it.

This paper is concerned with radially symmetric solutions of the Keller-Segel system with nonlinear signal production, as given by {u(t) = Delta(u) - del . (u del v), 0 = Delta v - mu(t) + f (u), mu(t) := 1/vertical bar Omega vertical bar integral(Omega)integral(u(., t)), in the ball Omega = B-R(0) subset of R-n for n >= 1 and R > 0, where f is a suitably regular function generalizing the prototype determined by the choice f (u) = u(kappa), u >= 0, with kappa > 0. The main results assert that if in this setting the number kappa satisfies kappa > 2/n, (star) then for any prescribed mass level m > 0, there exist initial data u0 with integral(Omega) u(0) = m, for which the solution of the corresponding Neumann initial-boundary value problem blows up in finite time. The condition in (star) is essentially optimal and is indicated by a complementary result according to which in the case kappa < 2/n, for widely arbitrary initial data, a global bounded classical solution can always be found.

Multistability is a ubiquitous feature in systems of geophysical relevance and provides key challenges for our ability to predict a system's response to perturbations. Near critical transitions small causes can lead to large effects and-for all practical purposes-irreversible changes in the properties of the system. As is well known, the Earth climate is multistable: present astronomical and astrophysical conditions support two stable regimes, the warm climate we live in, and a snowball climate characterized by global glaciation. We first provide an overview of methods and ideas relevant for studying the climate response to forcings and focus on the properties of critical transitions in the context of both stochastic and deterministic dynamics, and assess strengths and weaknesses of simplified approaches to the problem. Following an idea developed by Eckhardt and collaborators for the investigation of multistable turbulent fluid dynamical systems, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. The edge state attracts initial conditions belonging to such a boundary and, while being defined by the deterministic dynamics, is the gate facilitating noise-induced transitions between competing attractors. We use a simplified yet Earth-like intermediate complexity climate model constructed by coupling a primitive equations model of the atmosphere with a simple diffusive ocean. We refer to the climatic edge states as Melancholia states and provide an extensive analysis of their features. We study their dynamics, their symmetry properties, and we follow a complex set of bifurcations. We find situations where the Melancholia state has chaotic dynamics. In these cases, we have that the basin boundary between the two basins of attraction is a strange geometric set with a nearly zero codimension, and relate this feature to the time scale separation between instabilities occurring on weather and climatic time scales. We also discover a new stable climatic state that is similar to a Melancholia state and is characterized by non-trivial symmetry properties.

This paper deals with the Neumann problem for the coupled chemotaxis-haptotaxis model of cancer invasion given by [GRAPHICS] for x is an element of Omega and t > 0 with, respectively, given nonnegative initial data u(0) and w(0), where chi, xi and mu are positive parameters, Omega is a bounded domain in R-n, and n >= 1, with smooth boundary. The goal of this work is to identify two parallels between the solution behaviour in (star) and that in the corresponding two-component chemotaxis system obtained when w = 0: For the latter, it has been known that for any choice of u(0) is an element of C-0 (Omega) over bar, solutions are global and remain bounded under the condition [GRAPHICS] The first result of this paper says that the above statement remains true for arbitrarily large haptotactic ingredients: If (star star) holds, then for any xi > 0 and all reasonably smooth w(0), (star) possesses a globally defined solution which is bounded in each of its components. With regard to the qualitative solution behaviour, this work identifies an explicit smallness condition on w(0) which under the assumption (star star) asserts exponential decay of w in the large time limit, whereas both u and v persist in a certain sense.

In this paper, we study a reaction-diffusion vector-host epidemic model. We define the basic reproduction number R-0 and show that R-0 is a threshold parameter: if R-0 1 the model has a unique globally stable positive equilibrium. Our proof combines arguments from monotone dynamical system theory, persistence theory, and the theory of asymptotically autonomous semiflows.

We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrodinger (NLS) equation with the initial condition in the form of a rectangular barrier (a 'box'). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains-the dispersive dam break flows-generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.

In recent years the theory of the Wasserstein metric has opened up new treatments of diffusion equations as gradient systems, where the free energy or entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction-diffusion systems with reversible mass-action kinetic. The metric is constructed using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential. The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in Albinus et al (2002 Nonlinearity 15 367-83), which stimulated this work, have a gradient structure.