Showing papers in "Nonlinear Analysis-real World Applications in 2014"

A free boundary problem of a predator–prey model with higher dimension and heterogeneous environment

Abstract: This paper is concerned with a free boundary problem for a prey–predator model in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of an invasive or new predator species in which the free boundary represents the spreading front of the predator species and is described by Stefan-like condition. For simplicity, we assume that the environment and solutions are radially symmetric. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as t → ∞ and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The criteria for spreading and vanishing are given.

Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Abstract: Motivated by applications to various semilinear and quasi-linear stochastic partial differential equations (SPDEs) appeared in real world models, we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Levy processes. We illustrate the main results of our paper by showing how it can be applied to a large class of SPDEs such as stochastic reaction–diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

Dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response

Abstract: The purpose of this paper is to study the dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response. First, we analyze the well-posedness of the initial value problem of the model in the bounded domain Ω ⊆ R n . Then, we define the basic reproduction number R 0 which serves as a threshold to predict whether epidemics will spread, and by analyzing the corresponding characteristic equations of the uninfected steady state and infected steady state, respectively, we discuss the local stability of them. Moreover, by employing two Lyapunov functionals, we investigate the global stability of the two steady states. Finally, applying a known result, we show that there exist traveling wave solutions connecting the two steady states when R 0 > 1 , and there do not exist traveling wave solutions connecting the uninfected steady state itself when R 0 1 . Numerical simulations are provided to illustrate the main results.

Two spatial dimensional N-rogue waves and their dynamics in Mel’nikov equation

Abstract: By means of the Hirota bilinear method, explicit representations of general rogue waves for the Mel’nikov equation are explored in terms of determinants. As applications, it is found that this system admits bright- and dark-types rogue waves localized in two dimensional space. Furthermore, the superposition of such bright rogue waves are investigated graphically by different choices of the free parameters.

Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment

Abstract: Anti-retroviral treatments (ART) such as HAART have been used to control the replication of HIV virus in HIV-positive patients. In this paper, we study an in-host model of HIV infection with ART and carry out mathematical analysis of the global dynamics and bifurcations of the model in different parameter regimes. Among our discoveries is a parameter region for which backward bifurcation can occur. Biologically, the catastrophic behaviors associated with backward bifurcations may explain the sudden rebound of HIV viral load when ART is stopped, and possibly provide an explanation for the viral blips during ART suppression of HIV.

Nonlinear stability of flock solutions in second-order swarming models

Abstract: In this paper we consider interacting particle systems which are frequently used to model collective behaviour in animal swarms and other applications. We study the stability of orientationally aligned formations called flock solutions, one of the typical patterns emerging from such dynamics. We provide an analysis showing that the nonlinear stability of flocks in second-order models entirely depends on the linear stability of the first-order aggregation equation. Flocks are shown to be nonlinearly stable as a family of states under reasonable assumptions on the interaction potential. Furthermore, it is tested numerically that commonly used potentials satisfy these hypotheses and the nonlinear stability of flocks is investigated by an extensive case study of uniform perturbations.

Global dynamics of a virus dynamical model with general incidence rate and cure rate

Abstract: A virus dynamical model with general incidence rate and cure rate is proposed and analyzed. The system always admits a virus free equilibrium, which is shown to be globally asymptotically stable if the basic reproduction number R 0 ⩽ 1 by using the method of Lyapunov function. And there is a unique endemic equilibrium, which is locally asymptotically stable, if R 0 > 1 . Further, its global asymptotic stability is established by ruling out periodic solutions and using the Poincare–Bendixson property for three dimensional competitive systems. The model and mathematical results in [K. Hattaf, N. Yousfi, A. Tridan, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. RWA 13 (2012) 1866–1872] are generalized.

Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence

Abstract: We explore the homogenization limit and rigorously derive upscaled equations for a microscopic reaction–diffusion system modeling sulfate corrosion in sewer pipes made of concrete. The system, defined in a periodically-perforated domain, is semi-linear, partially dissipative and weakly coupled via a non-linear ordinary differential equation posed on the solid–water interface at the pore level. First, we show the well-posedness of the microscopic model. We then apply homogenization techniques based on two-scale convergence for a uniformly periodic domain and derive upscaled equations together with explicit formulas for the effective diffusion coefficients and reaction constants. We use a boundary unfolding method to pass to the homogenization limit in the non-linear ordinary differential equation. Finally, we give the strong formulation of the upscaled system.

Bifurcations and global dynamics in a predator-prey model with a strong Allee effect on the prey, and a ratio-dependent functional response

Abstract: We extend a previous study of a predator–prey model with strong Allee effect on the prey in which the functional response is a function of the ratio of prey to predator. We prove that the solutions are always bounded and non-negative, and that the species can always tend to long-term extinction. By means of bifurcation analysis and advanced numerical techniques for the computation of invariant manifolds of equilibria, we explain the consequences of the (dis)appearance of limit cycles, homoclinic orbits, and heteroclinic connections in the global arrangement of the phase plane near a Bogdanov–Takens bifurcation. In particular, we find that the Allee threshold in the two-dimensional system is given as the boundary of the basin of attraction of an attracting positive equilibrium, and determine conditions for the mutual extinction or survival of the populations.

Predictor homotopy analysis method: Two points second order boundary value problems

Abstract: The purpose of the present paper is to give a proof for a systematic method namely, predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of two points nonlinear second order boundary value problems of type u ″ = f ( x , u , u ′ ) . We present briefly the method first, and then some theorems are given to clarify this issue that, how one can predict the existence of the multiple solutions and obtain them simultaneously as well.

New results on the positive almost periodic solutions for a model of hematopoiesis

Abstract: This paper is concerned with a non-autonomous delayed model of hematopoiesis which is defined on the non-negative function space. Based on the definition of an almost periodic function, we employ a novel argument to establish a delay-independent criteria ensuring the existence, uniqueness, and global exponential stability of positive almost periodic solutions of the model with almost periodic coefficients and delays. Moreover, an example and its numerical simulation are given to illustrate the theoretical results.

Stability and turing instability in a diffusive predator–prey system with Beddington–DeAngelis functional response

Abstract: A diffusive two-species predator–prey system with the Beddington–DeAngelis functional response and subject to homogeneous Neumann boundary conditions is considered. In the region of parameters where the positive constant steady state is globally asymptotically stable when there exists no diffusion, the impact of the diffusion on the stability is analyzed in detail. The global asymptotic stability of the positive constant steady state is also considered by means of the upper and lower solutions method and the monotone iteration principle.

On a cross-diffusion segregation problem arising from a model of interacting particles

Abstract: We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.

Periodic solutions of Liénard equation with a singularity and a deviating argument

Abstract: In this paper, we study the existence of periodic solutions of the Lienard equation with a singularity and a deviating argument x ″ + f ( x ) x ′ + g ( t , x ( t − σ ) ) = 0 . When g has a strong singularity at x = 0 and satisfies a new small force condition at x = ∞ , we prove that the given equation has at least one positive T -periodic solution.

Dynamical complexity induced by Allee effect in a predator–prey model

Abstract: In this paper, we investigate the complex dynamics induced by Allee effect in a predator–prey model. For the non-spatial model, Allee effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee effect, the model is bistable. For the spatial model, without Allee effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee effect is not simple, but rich and complex.

Adaptive dynamics for a non-autonomous Lotka–Volterra model with size-selective disturbance☆

Abstract: This paper intends to develop a theoretical framework for investigating the evolutionary impact of size-selective disturbance on an evolving trait of individuals in a non-autonomous Lotka–Volterra competition model. We first construct an invasion fitness function, which involves the average growth rate and settles in a nonequilibrium attractor. Then using methods of adaptive dynamics and critical function analysis we investigate the evolution of body size related traits in a competitive community, and when having size-selective disturbance we obtain the conditions for continuously stable strategy and evolutionary branching. Our results show that (1) heavy harvesting can lead to rapidly stable evolution towards smaller body size, but planting has an opposite effect; (2) smaller harvest pressure can give rise to high levels of polymorphism during evolution; (3) planting can make for evolutionary branching, while harvesting can go against evolutionary branching and promote evolutionary stability. Thus we can conclude, from an evolutionary point of view, that planting can promote species diversity and spatial differentiation among populations; however, harvesting does the opposite.

Remarks on regularity criteria for the Navier–Stokes equations via one velocity component

Abstract: Recently, Pokorný and Zhou proved that if ‖ u 3 ‖ L ∞ ( 0 , T ; L 10 3 ( R 3 ) ) ≪ 1 or ‖ ∇ u 3 ‖ L ∞ ( 0 , T ; L 30 19 ( R 3 ) ) ≪ 1 , then the weak solution u of the 3D Navier–Stokes equations is regular on R 3 × ( 0 , T ] . In this paper we remove the smallness assumptions by using a new approach which may be of independent interest and further application.

Global well-posedness of the three dimensional magnetohydrodynamics equations

Abstract: In this paper we investigate the Cauchy problem for the incompressible MHD equations in three space dimensions. If the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients, global existence of mild solutions is obtained.

Asymptotic structure of the attractor for processes on time-dependent spaces

Abstract: We compare the asymptotic structure of the time-dependent attractor A t generated by the partial differential equation e u t t + α u t − Δ u + f ( u ) = g , where the positive function e = e ( t ) tends to zero as t → ∞ , with the global attractor A ∞ of its formal limit α u t − Δ u + f ( u ) = g . We establish an abstract result and we apply it to the proof of the convergence A t → A ∞ .

Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem

Abstract: In this work a nonlinear eigenvalue problem for a nonlinear autonomous ordinary differential equation of the second order is considered. This problem describes the process of propagation of transverse-electric electromagnetic waves along a plane dielectric waveguide with nonlinear permittivity. We demonstrate, as far as we know, a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues. The method is based on a simple idea that the distance between zeros of a periodic solution to the differential equation is the same for the adjacent zeros. This method has no connections with the perturbation theory or the notion of a bifurcation point. Theorem of equivalence between the eigenvalue problem and the dispersion equation is proved. Periodicity of the eigenfunctions is proved, a formula for the period is found, and zeros of the eigenfunctions are determined. The formula for the distance between adjacent zeros of any eigenfunction is given. Also theorems of existence and localization of the eigenvalues are proved.

Local well-posedness for the chemotaxis-Navier–Stokes equations in Besov spaces

Abstract: In this paper, we are concerned with the local well-posedness for the chemotaxis-Navier–Stokes equations in R d , d = 2 , 3 . By fully using the advantage of weighted function generated by heat kernel and Fourier localization technique, we obtain the existence and uniqueness of smooth solutions in Besov spaces. More importantly, we show a Beale–Kato–Majda type blow-up criterion with the help of a logarithmic inequality.

Weighted and well-balanced anisotropic diffusion scheme for image denoising and restoration

Abstract: Anisotropic diffusion is a key concept in digital image denoising and restoration. To improve the anisotropic diffusion based schemes and to avoid the well-known drawbacks such as edge blurring and ‘staircasing’ artifacts, in this paper, we consider a class of weighted anisotropic diffusion partial differential equations (PDEs). By considering an adaptive parameter within the usual divergence process, we retain the powerful denoising capability of anisotropic diffusion PDE without any oscillating artifacts. A well-balanced flow version of the proposed scheme is considered which adds an adaptive fidelity term to the usual diffusion term. The scheme is general, in the sense that, different diffusion coefficient functions can be utilized according to the need and imaging modality. To illustrate the advantage of the proposed methodology, we provide some examples, which are applied in restoring noisy synthetic and real digital images. A comparison study with other anisotropic diffusion based schemes highlight the superiority of the proposed scheme.

The existence of a global solution for one dimensional compressible viscous micropolar fluid with non-homogeneous boundary conditions for temperature

Abstract: We consider non-stationary 1-D flow of a compressible viscous heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are introduced. This problem has a unique generalized solution locally in time. With the help of this result and using the principle of extension we prove a global-in-time existence theorem.

Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on RN

Abstract: In this paper, we study the following quasi-linear elliptic equations of the form − ∑ i , j = 1 N D j ( a i , j ( x , u ) D i u ) + 1 2 ∑ i , j = 1 N D s a i , j ( x , u ) D i u D j u + V ( x ) u = h ( x , u ) , x ∈ R N , where h ∈ C ( R N × R , R ) , N ≥ 3 and V ∈ C ( R N , R ) . Under appropriate assumptions on V ( x ) and h ( x , u ) , some existence results for positive solutions, negative solutions and a sequence of high energy solutions are obtained via the perturbation method.

Limit cycles in discontinuous classical Liénard equations

Abstract: We study the number of limit cycles which can bifurcate from the periodic orbits of a linear center perturbed by nonlinear functions inside the class of all classical polynomial Lienard differential equations allowing discontinuities. In particular our results show that for any n ≥ 1 there are differential equations of the form x + f ( x ) x + x + s g n ( x ) g ( x ) = 0 , with f and g polynomials of degree n and 1 respectively, having [ n / 2 ] + 1 limit cycles, where [ ⋅ ] denotes the integer part function.

Bifurcation for a free boundary problem modeling tumor growth with inhibitors

Abstract: This paper deals with a free boundary problem modeling tumor growth with inhibitors. This problem has a unique radially symmetric stationary solution with radius r = R s . The tumor aggressiveness is modeled by a positive tumor aggressiveness parameter μ . It is shown that there exist a positive integer m ∗ ∗ ∈ R and a sequence of μ m , such that for each μ m ( m > m ∗ ∗ ) , symmetry-breaking solutions bifurcate from the radially symmetric stationary solutions.

Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction–diffusion equations

Abstract: We discuss free boundary problems modeling the diffusion of invasive or new species in a multi-dimensional ball or annulus, where unknown functions are population density and the outer boundary representing the spreading front of the species. We define spreading and vanishing to describe the asymptotic behaviors of radially symmetric solutions. The main purpose is to study the underlying principle to determine spreading and vanishing for the free boundary problems of general reaction–diffusion equations. We also focus on the problems with a logistic or bistable reaction term to show dichotomy results, vanishing speed and sufficient conditions for spreading or vanishing.

Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion

Abstract: In this paper we study a model of thermal explosion which is described by positive solutions to the boundary value problem { − Δ u = λ f ( u ) , x ∈ Ω , n ⋅ ∇ u + c ( u ) u = 0 , x ∈ ∂ Ω , where f , c : [ 0 , ∞ ) → ( 0 , ∞ ) are C 1 and C 1 , γ non decreasing functions satisfying lim u → ∞ f ( u ) u = 0 , Ω is a bounded domain in R N with smooth boundary ∂ Ω and λ > 0 is a parameter. Using the method of sub and super-solutions we show that the solution of this problem is unique for large and small values of parameter λ , whereas for intermediate values of λ solutions are multiple provided nonlinearity f satisfies some natural assumptions. An example of such nonlinearity which is most relevant to applications and satisfies all our hypotheses is f ( u ) = exp [ α u α + u ] for α ≫ 1 .

Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics

Abstract: We show the existence of a global weak dissipative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line with nonvanishing and distinct spatial asymptotics. The influence from the second component in the 2CH system on the regularity of the solution, and, in particular, the consequences for wave breaking, is discussed. Furthermore, the interplay between dissipative and conservative solutions is treated.

A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity

Abstract: In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which includes the thermal effects and considers the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using recent results from the theory of hemivariational inequalities and a fixed point argument.