Nontrivial Stationary Solutions Research Articles

A meeting point of entropy and bifurcations in cross-diffusion herding

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

Inverted pendulum state of a polariton Rabi oscillator

Exciton-photon beats known as polariton Rabi oscillations in semiconductor microcavities are usually excited by short pulses of light. We consider a different pumping scheme, assuming a cw pumping of the Rabi oscillator from an exciton reservoir. We account for the initial pulse of light setting the phase, exciton decay due to exciton-phonon and exciton-exciton scattering, photon leakage, and blueshift of the exciton resonance due to interactions. We find non-trivial stationary solutions reminiscent of Kapitza pendulum, where polaritons are accumulated at the upper branch while the lower branch empties.

Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

Infinite energy solutions to inelastic homogeneous Boltzmann equations

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a class of spatially homogeneous kinetic equations in space dimensions $d\ge2$. Our results cover in particular Bobylev's model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index $\alpha\in(0,2)$ such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $\alpha$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation. Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $\alpha$-stable distribution converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of projections of randomly rotated i.i.d. random vectors.

Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet, Neumann or Robin boundary conditions imposed on ψ at infinity. This class includes the regular Kerr–AdS black holes satisfying the Hawking–Reall bound r+2>|a|l. We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of ψ and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr–AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions.

COMPETING EFFECTS OF ATTRACTION VS. REPULSION IN CHEMOTAXIS

We consider the attraction–repulsion chemotaxis system [Formula: see text] under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝn with smooth boundary, where χ ≥ 0, ξ ≥ 0, α > 0, β > 0, γ > 0, δ > 0 and τ = 0, 1. We study the global solvability, boundedness, blow-up, existence of non-trivial stationary solutions and asymptotic behavior of the system for various ranges of parameter values. Particularly, we prove that the system with τ = 0 is globally well-posed in high dimensions if repulsion prevails over attraction in the sense that ξγ - χα > 0, and that the system with τ = 1 is globally well-posed in two dimensions if repulsion dominates over attraction in the sense that ξγ - χα > 0 and β = δ. Hence our results confirm that the attraction–repulsion is a plausible mechanism to regularize the classical Keller–Segel model whose solution may blow up in higher dimensions.

Nonclassical analysis of the nonlinear Kompaneets equation

The dimensional nonlinear Kompaneets (NLK) equation \({u_{t} = x^{-2}[x^{4}(\alpha{u}_{x} + \beta{u} + \gamma{u}^{2})]_{x}}\) describes the spectra of photons interacting with a rarefied electron gas. Recently, Ibragimov obtained some time-dependent exact solutions for several approximations of this equation. In this paper, we use the nonclassical method to construct time-dependent exact solutions for the NLK equation \({u_{t} = x^{-2}[x^{4}(\alpha{u}_{x} + \gamma{u}^{2})]_{x}}\) for arbitrary constants α > 0, γ > 0. Solutions arising from “nonclassical symmetries” are shown to yield wider classes of time-dependent exact solutions for the NLK equation \({u_{t} = x^{-2}[x^{4}(\alpha{u}_{x} + \gamma{u}^{2})]_{x}}\) beyond those obtained by Ibragimov. In particular, for five classes of initial conditions, each involving two parameters, previously unknown explicit time-dependent solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. Three of the classes exhibit quiescent behavior, i.e., \({\lim_{t\rightarrow \infty}u(x,t) = 0}\) , and the other two classes exhibit blow-up behavior in finite time. As a consequence, it is shown that the corresponding nontrivial stationary solutions are unstable.

Reaction diffusion equations and the chronification of liver infections

A predator-prey system is used to model the time-dependent virus and lymphocyte population during a liver infection. We show mathematically that the resulting reaction-diffusion equation has non-trivial stationary solutions whenever the underlying domain is sufficiently large or fissured. The non-trivial stationary solutions are interpreted as chronic liver infections. Thus qualitative differences between acute and chronic hepatitis infections become dispensable. Finally, numerical simulations for the chronification are presented.

Propagation and Blocking in Periodically Hostile Environments

We study the persistence and propagation (or blocking) phenomena for a species in periodically hostile environments. The problem is described by a reaction–diffusion equation with the zero Dirichlet boundary condition. We first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large-time behavior of the solution of the initial boundary value problem. In addition to the main goal, we then study a sequence of approximated problems in the whole space with reaction terms with very negative growth rates which are outside the domain under investigation. Finally, for a given unit vector, by using the information of the minimal speeds of approximated problems, we provide a simple geometric condition for the blocking of propagation and we derive the asymptotic behavior of the approximated pulsating travelling fronts. Moreover, for the case of the constant diffusion matrix, we provide two conditions for which the limit of approximated minimal speeds is positive.

Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion

We prove the existence of global minimizers of a class of free energies related to aggregation equations with degenerate diffusion on R d . Such equations arise in mathematical biology as models for organism group dynamics which account for competition between the tendency to aggregate into groups and nonlinear diffusion to avoid overcrowding. The existence of non-trivial stationary solutions with minimal energy representing coherent groups in R d is therefore of interest. A scaling criticality that measures the balance between the diffusive and aggregative forces as mass spreads is shown to govern the existence and non-existence of global minimizers. The primary difficulty confronted here is the inability to verify strict subadditivity conditions for biologically relevant problems which violate homogeneity-type assumptions known to be sufficient. To recover, we show that sufficiently degenerate diffusion provides a weaker condition from which tightness of symmetrized infimizing sequences can be recovered, even when the nonlocal attractive force is extremely weak.

Bifurcation analysis to the Lugiato–Lefever equation in one space dimension

We study the stability and bifurcation of steady states for a certain kind of damped driven nonlinear Schrödinger equation with cubic nonlinearity and a detuning term in one space dimension, mathematically in a rigorous sense. It is known by numerical simulations that the system shows lots of coexisting spatially localized structures as a result of subcritical bifurcation. Since the equation does not have a variational structure, unlike the conservative case, we cannot apply a variational method for capturing the ground state. Hence, we analyze the equation from a viewpoint of bifurcation theory. In the case of a finite interval, we prove the fold bifurcation of nontrivial stationary solutions around the codimension two bifurcation point of the trivial equilibrium by exact computation of a fifth-order expansion on a center manifold reduction. In addition, we analyze the steady-state mode interaction and prove the bifurcation of mixed-mode solutions, which will be a germ of localized structures on a finite interval. Finally, we study the corresponding problem on the entire real line by use of spatial dynamics. We obtain a small dissipative soliton bifurcated adequately from the trivial equilibrium.

Analysis of non-linear Klein–Gordon equations using Lie symmetry

This work obtains the stationary solutions of the non-linear Klein–Gordon equations in 1+1 dimensions. The technique that is used to carry out the analysis is the Lie symmetry approach. There are five types of non-linearity that are studied in this work. In each case, the analysis yields non-trivial stationary solutions; it is the first time that this has been seen.

Multiple stable patterns in a balanced bistable equation with heterogeneous environments

In this article, we present a simple approach to show the existence of nontrivial stable stationary solutions to a balanced bistable reaction–diffusion equation with heterogeneous environments.

Pathwise stationary solutions for stochastic neural networks with delay

In this paper, a nontrivial stationary solution for a stochastic neural network with delay is studied. The analysis is done in the context of the theory of random dynamical system and the idea of M-matrices.

Convection in a Two-Layer System with a Deformable Interface Under Low Gravity Conditions

The onset of thermal convection in a system of two horizontal layers of immiscible liquids of similar densities is studied under low gravity conditions. A constant heat flux is prescribed at both rigid boundaries. A generalized Boussinesq approach that allows correct accounting for the interface deformation is used. The long-wave perturbations emerge under low-gravity conditions; either monotonic or oscillatory modes are critical depending on the problem. Moreover, two different modes of the monotonic instability exist. For the first instability mode, the convection dominates, whereas the interface remains nearly undeformable. The second monotonic instability mode is substantially related to interface deformations. The system of non-linear amplitude equations describing the behavior of long-wave regimes at finite-amplitude interface deflection and finite supercriticalities is obtained. The analytical and numerical investigations of these equations show that the stable non-trivial stationary solutions are absent, and after a transient at least one of the layers is split into the areas not connected to each other. The nonlinear regimes of cellular convection are studied numerically by the Level Set method.

Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry

In this paper we study the existence of stationary solutions for the following discrete vector nonlinear Schrödinger equation i ∂ ϕ n ∂ t = - Δ ϕ n + τ n ϕ n - J f ( n , | ϕ n | ) ϕ n , where ϕ n is a sequence of 2-component vector, J = 0 1 1 0 , Δ ϕ n = ϕ n + 1 + ϕ n - 1 - 2 ϕ n is the discrete Laplacian in one spatial dimension and sequence τ n is assumed to be N -periodic in n , i.e. τ n + N = τ n . We prove the existence of infinitely many nontrivial stationary solutions for this system by variational methods. The same method can also be applied to obtain infinitely many breather solutions for single discrete nonlinear Schrödinger equation.

Stationary solutions to the Falk system on shape memory alloys

AbstractWe study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physically closed stationary state is formulated as a nonlinear eigenvalue problem with a non‐local term. Then, some results on existence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamically stable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.

Convergence rates for the compressible Navier-Stokes equations with general forces

For the viscous and heat-conductive fluids governed by the compressible Navier-Stokes equations with external force of general form in ℝ 3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H 3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H 3 and bounded in L 6/5.

Ferrofluid patterns in a radial magnetic field: Linear stability, nonlinear dynamics, and exact solutions

The response of a ferrofluid droplet to a radial magnetic field is investigated, when the droplet is confined in a Hele-Shaw cell. We study how the stability properties of the interface and the shape of the emerging patterns react to the action of the magnetic field. At early linear stages, it is found that the radial field is destabilizing and determines the growth of fingering structures at the interface. In the weakly nonlinear regime, we have verified that the magnetic field favors the formation of peaked patterned structures that tend to become sharper and sharper as the magnitude of the magnetic effects is increased. A more detailed account of the pattern morphology is provided by the determination of nontrivial exact stationary solutions for the problem with finite surface tension. These solutions are obtained analytically and reveal the development of interesting polygon-shaped and starfishlike patterns. For sufficiently large applied fields or magnetic susceptibilities, pinch-off phenomena are detected, tending to occur near the fingertips. We have found that the morphological features obtained from the exact solutions are consistent with our linear and weakly nonlinear predictions. By contrasting the exact solutions for ferrofluids under radial field with those obtained for rotating Hele-Shaw flows with ordinary nonmagnetic fluids, we deduce that they coincide in the limit of very small susceptibilities.

Representation formulas for some 1-dimensional linearized eigenvalue problems

We study nontrivial stationary solutions to a nonlinear boundary value problem with parameter $\varepsilon>0$ and the corresponding linearized eigenvalue problem. By using a particular solution of a linear ordinary differential equation of the third order, we give expressions of all eigenvalues and eigenfunctions to the linearized problems. They are completely determined by a characteristic function which consists of complete elliptic integrals. We also show asymptotic formulas of eigenvalues with respect to sufficiently small $\varepsilon$. These results give important information for profiles of corresponding eigenfunctions.