Uniform Boundedness Of Solutions Research Articles

Finite-time blow-up of weak solutions to a chemotaxis system with gradient dependent chemotactic sensitivity

This paper deals with the parabolic–elliptic chemotaxis system with gradient dependent chemotactic sensitivity,{ut=Δu−χ∇⋅(u|∇v|p−2∇v),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0, where Ω:=BR(0)⊂Rn(n≥2) is a ball with some R>0, χ>0, p∈(nn−1,∞) and μ:=1|Ω|∫Ωu0, where u0 is an initial datum of arbitrary size. In the case that p∈(1,nn−1), Negreanu and Tello (2018) [23] established global existence and uniform boundedness of solutions, whereas when p∈(nn−1,2), Tello (2022) [31] showed that solutions blow up in finite time under the condition that μ>6 and χ is large enough. These works imply that the number p=nn−1 certainly plays the role of a critical blow-up exponent, and it is expected that when p>nn−1, for arbitrary μ>0 the system admits at least one solution which blows up in finite time. The purpose of this paper is to prove that this conjecture is true within a framework of weak solutions with a moment inequality.

Inverse problems for nonlinear quasi-variational hemivariational inequalities with application to obstacle problems of elliptic type

In this paper we study an inverse problem governed by a constrained nonlinear elliptic quasi-variational hemivariational inequality. The inequality involves two nondifferentiable functions which directly depend on solutions. We solve the direct problem and obtain the properties of weak closedness and uniform boundedness of solutions, which develops some existing results in literature. Then, using a regularized optimization method, we deal with an inverse problem to seek the identification of parameter functions appearing in the leading operator and constraint set. Finally, we apply the abstract result to a concrete inverse problem of identifying the parameter and obstacle functions in elliptic obstacle problems with the p-Laplace operator and mixed boundary conditions.

Spreading Speed in the Fisher-KPP Equation with Nonlocal Delay

This paper is concerned with the Fisher-KPP equation with diffusion and nonlocal delay. Firstly, we establish the global existence and uniform boundedness of solutions to the Cauchy problem. Then, we establish the spreading speed for the solutions with compactly supported initial data. Finally, we investigate the long time behavior of solutions by numerical simulations.

Uniform Boundedness of Solutions for a Two Species Taxis System with Intraspecific and Interspecific Competition

In this paper we study a nonlinear system of reaction–diffusion differential equations consisting of an ordinary equation coupled to a fully parabolic chemotaxis system. This system constitutes a mathematical model for the evolution of two populations that are competing for a common resource, one of which is subject to chemotaxis. Under suitable assumptions we prove the global in time existence and the boundedness of the classical solutions of this system in a two-dimensional bounded domain. In addition, the asymptotic behavior of the solutions for a particular case of the problem data is obtained.

The effect of the force of infection and treatment on the disease dynamics of an SIS epidemic model with immigrants

A basic simple SIS epidemic model is proposed and analyzed. The population is infected by a transmissible disease. The uniform boundedness of solutions is established. Local and global stability dynamics of the systems are discussed. The basic model is modified by incorporating a restricted supply of treatment and immigrants considering that a fraction of the immigrants arrive with the same infection into the system. We have calculated the basic reproduction numbers for the proposed model systems which play an important role in the existence of endemic equilibria. Through theoretical analysis, we have shown that the disease-free equilibrium point of the modified system possesses a transcritical bifurcation. An endemic equilibrium state exhibits a Hopf bifurcation. Sotomayor’s theorem on local bifurcation theory yields the explicit formula determining the characteristic of the bifurcating periodic solutions. A significant conclusion is that the force of infection and treatment have an opposite effect on the stability dynamics of the proposed SIS model system. The study reveals that the difference between the maximum treatment capacity and the infected immigrants plays a significant role on the disease endemicity of the epidemic system. Numerical simulations are presented to validate the analytical findings.

ANALYSIS OF A REACTION–DIFFUSION EPIDEMIC MODEL WITH ASYMPTOMATIC TRANSMISSION

We develop a dynamic model of an evolving epidemic in a spatially inhomogeneous environment. We analyze the model, as a system of reaction–diffusion partial differential equations, to predict the outbreak and spatio-temporal spread of the disease. The model features both an asymptomatic infectious stage and symptomatic infectious stage, with both the asymptomatic and the symptomatic infected populations dispersing through the susceptible population. We prove the existence and uniform boundedness of solutions, and investigate their long-time behavior. We apply the spatially homogeneous version of the model to the current COVID-19 epidemic in Brazil.

Dynamics of a delayed predator–prey interaction incorporating nonlinear prey refuge under the influence of fear effect and additional food

The main object of this work is to analyse the impacts of fear effect on the dynamics of predator–prey interaction incorporating non linear prey refuge to control the extinction of predator. We have taken a Holling type II functional response in presence of additional food. The positivity and uniform boundedness of solutions have been discussed. The feasibility conditions of all equilibrium points and their stability behaviours are derived. Next, we have studied the existence of local bifurcation (transcritical bifurcation) of codimension 1. Moreover, impacts of fear effect have been studied analytically and noticed that the predator biomass can not only be decreased at the interior equilibrium by the effect of fear but the system also be stabilized by excluding the existence of periodic solutions through Hopf-bifurcation. In addition, uniform persistence of our proposed system has been discussed analytically. Further, we have analysed the stability behaviours of the interior equilibrium for all combinations of the delay factors (τ and τ′) and observed that the delay parameters may produce oscillating behaviours through a Hopf-bifurcation. Numerical simulations have been illustrated using MATLAB to validate all the analytical results. Numerically, the impact of coefficient of prey refuge on the predator population has also been performed.

A Mathematical Study of a Two Species Eco-Epidemiological Model with Different Predation Principles

This paper formulates and analyzes a predator-prey model with disease in the prey. Mathematical analysis of the model system concerns the existence, uniqueness and uniform boundedness of solutions in the positive octant. The threshold condition for epidemic and the conditions for persistence are obtained. Moreover, the system is analyzed for local stability, global stability around several equilibria. Hopf-bifurcation with its nature and the stability of the bifurcating limit cycle are studied around the disease free equilibrium point. Numerical simulations are performed to justify the analytical findings. Eco-epidemilogical significance and implications of the concluded results are discussed as well.

Analysis of a Reaction–Diffusion Cholera Model with Distinct Dispersal Rates in the Human Population

In this paper, we investigate the long term behaviors of a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition. The parameters in the model involving space are typical space-dependent due to spatial heterogeneity. We consider the case that the dispersal rates of the susceptible and infected hosts are different, no diffusion term in the cholera equation and bilinear incidence infection mechanism. The existence of global solution, uniform boundedness of solution, asymptotic smoothness of semiflows and existence of global attractor are also addressed. We define the basic reproduction number $$\mathfrak {R}_0$$ for the model for the disease transmission in spatially homogeneous environment and establish a threshold type result for the disease eradication or uniform persistence. Considering the cases that either the dispersal rate of the susceptible individuals or the dispersal rate of the infected individuals approaches zero, we investigate the asymptotical profiles of the endemic steady state. Our results suggest that: cholera can be eliminated by limiting the movement of the susceptible individuals, while limiting the mobility of the infected hosts, the infected individuals concentrate on certain points in some circumstances.

Mathematical analysis of toxin-phytoplankton-fish model with self-diffusion and cross-diffusion

In this paper we propose a nonlinear reaction-diffusion system describing the interaction between toxin-producing phytoplankton and fish population. We analyze the effect of self- and cross-diffusion on the dynamics of the system. The existence, uniqueness and uniform boundedness of solutions are established in the positive octant. The system is analyzed for various interesting dynamical behaviors which include boundedness, persistence, local stability, global stability around each equilibrium based on some conditions on self- and cross-diffusion coefficients. The analytical findings are verified by numerical simulation.

On a Cahn–Hilliard system with convection and dynamic boundary conditions

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn–Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn–Hilliard cases are investigated, and a number of results are proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo–Galerkin scheme, is introduced and rigorously discussed.

Mathematical analysis of a predator–prey eco-epidemiological system under the reproduction of infected prey

We have proposed and analyzed a mathematical model of a predator–prey system where prey is infected but able to reproduce logistically with predators’ linear functional response. The existence, uniqueness and uniform boundedness of solutions are established in the positive octant. The threshold condition for epidemic and the conditions for persistence of the model system are derived. Several equilibria states and their feasibility conditions are found. The system is analyzed for various interesting dynamical behaviors which include boundedness, persistence, local stability, global stability around each of the equilibria. We have also investigated that the system undergoes a Hopf-bifurcation around the disease free equilibrium point. We have studied the direction and stability of the Hopf-bifurcating periodic oscillation. The analytical findings are verified by numerical simulation.

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotactic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.

Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach

Let u μ, x, s (., 0) be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space X $$\left \{\begin{array}{ll}\dot{u}(t) = A(t)u(t) + e^{i\mu t}x, \quad t > s \\ u(s) = 0. \end{array} \right.$$ Here, x is a vector in X, μ is a real number, q is a positive real number and A(·) is a q-periodic linear operator valued function. Under some natural assumptions on the evolution family \({\mathcal{U} = \{U(t, s): t \geq s\}}\) generated by the family {A(t)}, we prove that if for each μ, each s ≥ 0 and every x the solution u μ, x, s (·, 0) is bounded on R + by a positive constant, depending only on x, then the family \({\mathcal{U}}\) is uniformly exponentially stable. The approach is based on the theory of evolution semigroups.

On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating arguments

This paper is devoted to the mathematical analysis of the stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions and some examples are given to illustrate the theoretical analysis in this work.

Global existence of solutions to reaction-diffusion systems without conditions on the nonlinearities growth

It is well known that, for reaction–diffusion systems, if the nonlinearities grow faster than a polynomial, nothing seems to be known for instance. The purpose of this paper is to give sufficient conditions guaranteeing global existence, uniqueness and uniform boundedness of solutions for coupled reaction–diffusion equations without condition growth on the reactions terms f and g in case f + g ≠ 0. These systems possess many and various applications in physics as the diffusion of the Phosphorus in the Silicone or some models describing some nuclear reactions; there have also been other applications in chemistry and biology. Our techniques are based on the Lyapunov functional methods. Copyright © 2010 John Wiley & Sons, Ltd.

Reaction-Diffusion Equations with Polynomial Drifts Driven by Fractional Brownian Motions

A reaction-diffusion equation on [0, 1] d with the heat conductivity κ > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to κ. Also we show that if κ tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation.

Boundedness of global solutions for a supercritical semilinear heat equation and its application

We consider a Cauchy problem and a Cauchy-Dirichlet problem in a ball in R N for a semilinear heat equation u t = Δu + u p with radially symmetric nonnegative initial data. Let p > N - 2N - 1 N - 4 - 2 N - 1 and N ≥ 11. It is proved that if u is a global solution of the Cauchy problem with initial data satisfying a condition near infinity, then there exists a positive constant C such that |u(t)|∞ < C for all t ≥ 0. The uniform boundedness of solutions for the Cauchy-Dirichlet problem in a ball is also given. The result is used to show the existence of a solution for the Cauchy-Dirichlet problem in a ball which exhibits the type II blowup.

On uniform boundedness of parameterized discrete-time systems with decaying inputs: applications to cascades

A framework for controller design of sampled-data nonlinear systems via their approximate discrete-time models has been established recently. Within this framework naturally arises the need to investigate stability properties of parameterized discrete-time systems. Further results that guarantee appropriate stability of the parameterized family of discrete-time systems that is used within this framework have been also established for systems with cascaded structure. A fundamental condition that is required in this framework is uniform boundedness of solutions of the cascade. However, this is difficult to check in general. In this paper we provide a range of sufficient conditions for uniform boundedness that are easier to check. These results further contribute to the toolbox for controller design of sampled-data nonlinear systems via their approximate discrete-time models.

Razumikhin technique for boundedness of the solutions of impulsive integrodifferential equations

This paper investigates uniform boundedness of solutions of nonlinear impulsive integrodifferential equations. The Razumikhin technique using piecewise continuous Lyapunov functions is applied. The boundedness in both cases of fixed and variable moments are studied. The obtained sufficient conditions significantly depend on the moments of impulses.