Nontrivial Steady-state Solutions Research Articles

Existence and stability of boundary spike layer solutions of an attractive chemotaxis model with singular sensitivity and nonlinear consumption rate of chemical stimuli

This paper is devoted to the study of the existence and stability of non-trivial steady state solutions to the following coupled system of PDEs on the half-line R+=(0,∞): ut=uxx−χ[u(lnw)x]x,wt=ɛwxx−uγwm, which is a model of chemotaxis of Keller–Segel type. When u is subject to the no-flux boundary condition, w equals a positive value at the origin, and assuming the functions vanish at the far field, a unique steady state (U,W) is constructed under suitable restrictions on the system parameters, which is capable of describing fundamental phenomena in chemotaxis, such as spatial aggregation. Moreover, the steady state is shown to be nonlinearly asymptotically stable if (u0−U) carries zero mass, w0(x) matches W(x) at the far field, and the initial perturbation is sufficiently small in weighted Sobolev spaces.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

Global existence, blow-up and dynamical behavior in a nonlocal parabolic problem with variational structure

In this paper, we study an initial boundary value problem for a nonlocal parabolic equation that includes a diffusion term and convex–concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, we prove the existence of global solutions and obtain some blow-up conditions. Given the variational structure of the problem, we utilize the Mountain-pass theorem to demonstrate the existence of nontrivial steady-state solutions. Additionally, we establish the dynamical behavior of global solutions whose trajectory is relatively compact in H01(Ω), and which uniformly converge to some non-zero steady state after a long time, owing to the energy functional satisfying the P.S. condition. Finally, we derive an unstable steady states sequence by using another minimax theorem.

A numerical model for predicting waves run-up on coastal areas

ABSTRACT In this study, numerical investigations are performed to validate a numerical model for the prediction of waves propagation and waves run-up on coastal zones. The proposed numerical model is based on the depth-averaged two-dimensional dispersive shallow water equations with source terms due to variable bottom topography and bed friction effects. We propose to solve the resulting nonlinear system using a well-balanced positivity preserving Godunov-type finite volume method on unstructured triangular grids. We used a semi-implicit scheme for the friction terms and a well-balanced formulation for the bottom topography. Moreover, we prove that the numerical scheme exactly preserves a class of nontrivial steady-state solutions. To validate the proposed numerical model against experiments, we first demonstrate its ability to preserve nontrivial steady-state solutions over a slanted surface and then we model several laboratory experiments for the prediction of waves run-up on sloping beaches. The numerical simulations are in good agreement with laboratory experiments which confirms the robustness and accuracy of the proposed numerical model in predicting waves propagation on coastal areas.

Hopf bifurcation in a spatial heterogeneous and nonlocal delayed reaction–diffusion equation

A spatial heterogeneous and nonlocal delayed reaction–diffusion equation is investigated. The existence of the nontrivial steady-state solution is shown by steady-state bifurcation at the trivial solution. The form of the positive spatially nonhomogeneous steady state is given under certain conditions. Based on the form, its stability and Hopf bifurcation are analyzed so that the periodic and the quasi-periodic solutions near the spatially nonhomogeneous steady state are demonstrated.

On a hyperbolic-parabolic chemotaxis system.

Stability of steady state solutions associated with initial and boundary value problems of a coupled fluid-reaction-diffusion system in one space dimension is analyzed. It is shown that under Dirichlet-Dirichlet type boundary conditions, non-trivial steady state solutions exist and are locally stable when the system parameters satisfy certain constraints.

Accumulation times for diffusion-mediated surface reactions

In this paper we consider a multiparticle version of a recent probabilistic framework for studying diffusion-mediated surface reactions. The basic idea of the probabilistic approach is to consider the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a totally reflecting boundary; the effects of surface reactions are then incorporated via an appropriate stopping condition for the local time. The propagator is determined by solving a Robin boundary value problem, in which the constant rate of reactivity is identified as the Laplace variable z conjugate to the local time, and then inverting the solution with respect to z. Here we reinterpret the propagator as a particle concentration in which surface absorption is counterbalanced by particle source terms. We investigate conditions under which there exists a non-trivial steady state solution, and analyze the relaxation to steady state by calculating the corresponding accumulation time. In particular, we show that the first two moments of the stopping local time density have to be finite.

Attraction–repulsion taxis mechanisms in a predator–prey model

We consider a predator–prey model where the predator population favors the prey through biased diffusion toward the prey density, while the prey population employs a chemical repulsive mechanism. This leads to a quasilinear parabolic system. We first establish the global existence of positive solutions. Thereafter we show the existence of nontrivial steady state solutions via bifurcation theory, then we discuss the stability of these branch solutions. Through numerical simulation we analyze the nature of patterns formed and interpret results in terms of the survival and distribution of the two populations.

Global behavior of a reaction-diffusion model with time delay and Dirichlet condition

This paper is concerned with analysis of the global behavior for a type of reaction-diffusion system incorporating the pathogens' incubation period, which describes the infection of bacteria in a population. By using the operator semigroup theory and dynamical system approaches, the global dynamics with respect to the homogeneous Dirichlet problem are investigated. Firstly, the positivity, boundedness, compactness, and smoothness of the semiflow are obtained. Then, we study the topology on the ω-limit set and the dynamical behaviors, including permanence, existence, uniqueness and attractiveness of nontrivial steady-state solutions. Finally, our main results are employed to characterize threshold dynamics for two special models with Rick type and Mackey-Glass type infectivity nonlinearities respectively.

Oscillation Phenomena of Rayleigh Equation Disturbed by Superposition of Two Harmonic Excitation

The Rayleigh Nonlinear Oscillation Equation Phenomenon disturbed by two external harmonic forces. One of which is small, will be discussed. Using the Multiple Scales method to obtain analytical solutions, several cases of resonance are determined. Steady-state solutions and their stabilities will be investigated. Depending on the value of amplitudes and frequencies of the external forces given, for a non-trivial steady-state solution, there are two possibilities, namely a periodic solution, and a non-periodic solution that raises the beating phenomenon.

The Dynamics of the Epidemic Process with the Antibiotic-Resistant Variant of the Pathogen Agent

The parasite–host system is considered with two different variants of a pathogen agent with complete cross protection, where one of the variants may transform into another with a certain probability. This system corresponds to the epidemic process initiated by the antibiotic-resistant variant of the pathogen agent. The behavior of solutions is studied. It is proved that the main case has a unique nontrivial steady-state solution that is a global attractor. The speed of the exponential approach of small deviations to the steady-state solution is obtained.

Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study

The nonlinear vibration behavior and dynamic instability of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads is the main objective of the present paper. Firstly, a short Euler–Bernoulli nanobeam is modeled and exposed to an external parametric excitation. Based on the nonlocal continuum theory and nonlinear von Karman beam theory, the nonlinear governing differential equation of motion is derived. Secondly, to transport the partial differential equation to the ordinary differential equation, Galerkin method is applied. Then, multiple scales method, as an analytical approach, is used to solve the equation. At the end, modulation equation of Euler–Bernoulli nanobeams is obtained. Then, to evaluate the dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are studied and investigated. As a result, it can be observed that the damping coefficient plays an effective role as well as parametric excitation in stability and frequency response of the system.

Parametrically excited nonlinear dynamic instability of reinforced piezoelectric nanoplates

The nonlinear dynamic instability of reinforced piezoelectric nanoplates exposed to a parametric excitation and an electric voltage is the objective of the present paper. Firstly, a piezoelectric nanoplate reinforced with two graphene layers and resting on a visco-elastic foundation is modeled. Secondly, the piezoelectric nonlocal elasticity theory, the Kelvin-Voigt model, von Karman nonlinear relations and Hamilton’s principle, respectively, are used to derive the nonlinear governing differential equation of motion. In the next step, to transform partial differential equation to ordinary one and then, solve the equation, the Galerkin technique and multiple time scales method are employed respectively. At the end, the modulation equation of reinforced piezoelectric nanoplates is obtained. Emphasizing the effect of the electric voltage and parametric excitation on dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. The main results emphasize that the damping coefficient is responsible of the bifurcation point variation, while the amplitude response depends on the term of natural frequency. Therefore, damping can have a strong influence on the system.

Parametric excitation of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads: Nonlinear vibration and dynamic instability

The nonlinear vibration behavior and dynamic instability of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads is the main objective of present paper. Firstly, a short Euler–Bernoulli nanobeam is modeled and exposed to an external parametric excitation. Based on the nonlocal continuum theory and nonlinear von Karman beam theory, the nonlinear governing differential equation of motion is derived. Secondly, to transport the partial differential equation to the ordinary differential equation, Galerkin method is applied. Then, multiple scales method, as an analytical approach, is used to solve the equation. At the end, modulation equation of Euler–Bernoulli nanobeams is obtained. Then, to evaluate the dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are studied and investigated. As a results, it can be observed that the damping coefficient plays an effective role as well as parametric excitation in stability and frequency response of the system.

Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier–Stokes Equations in High Dimensions

Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$. The result implies the nonuniqueness of finite energy weak solutions for the Navier-Stokes equations in dimensions $d \geq 4$. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.

Existence of a Nontrivial Steady-State Solution to a Parabolic-Parabolic Chemotaxis System with Singular Sensitivity

This paper establishes the existence of a nontrivial steady-state solution to a parabolic-parabolic coupled system with singular (or logarithmic) sensitivity and nonlinear source arising from chemotaxis. The proofs mainly rely on the maximum principle, the implicit function theorem, and the Hopf bifurcation theorem.

Reprint of: Residual equilibrium schemes for time dependent partial differential equations

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

Patterns in a nonlocal time-delayed reaction–diffusion equation

In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous) steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet/Neumann boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to population models with one-dimensional spatial domain.

Bifurcation of positive solutions to scalar reaction–diffusion equations with nonlinear boundary condition

The bifurcation of non-trivial steady state solutions of a scalar reaction–diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.

Large amplitude free vibration of a viscoelastic beam carrying a lumped mass–spring–damper

Numerous engineering structures can be modeled as flexible beams carrying appendages. Physical time-dependent phenomena in such structures appear by considering nonlinear models. Accordingly, the free vibration of a nonlinear Rayleigh beam carrying a linear mass–spring–damper is investigated in this research. The nonlinearity is due to the axial stress changes, and the viscoelastic characteristic of the beam is described by the Kelvin–Voigt model. The governing nonlinear equations of motion are derived by using Hamilton’s principle and solved by the multiple scales method. The presence of the linear damper in the mass–spring–damper system causes a contradiction in the time response of the mass obtained from the classical procedure of the multiple scales method. Therefore, the problem is solved by some manipulations in the process of the multiple scales method. Then, the complex-valued resonance frequencies and mode-shapes are derived by applying the method of power series with the aid of Green function concept. Considering the solvability condition, the nonlinear time-dependent resonance frequencies and the vibration response are obtained. The stability of the solution is studied by using the forward and pullback attractor ideas. Finally, the vibration of the system involving 3:1 internal resonance is investigated, and the stability boundaries of the trivial and non-trivial steady-state solutions are derived based on Lyapunov’s first method. It is shown that the main parameters of the MSD can provide 3:1 internal resonance condition.