Unique Global Solution Research Articles

Dynamics of a reaction–diffusion waterborne pathogen model with free boundaries

In this paper, we consider a reaction–diffusion waterborne pathogen model with free boundary. We first prove that this problem has a unique global solution, and then we show that its longtime behavior is determined by a spreading–vanishing dichotomy. Moreover, we obtain sharp criteria for spreading and vanishing, which is very different from the results in Zhou et al. (2018) that the epidemic will spread for the basic reproduction number is larger than 1.

On the global existence and analyticity of the mild solution for the fractional Porous medium equation

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained.

Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces

In this article, we prove the existence of a unique global solution for the critical case of the generalized Navier-Stokes equations in Lei-Lin and Lei-Lin-Gevrey spaces, by assuming that the initial data is small enough. Moreover, we obtain a unique local solution for the subcritical case of this system, for any initial data, in these same spaces. It is important to point out that our main result is obtained by discussing some properties of the solutions for the heat equation with fractional dissipation.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/78/abstr.html

Global existence of large solutions for the three‐dimensional incompressible Navier–Stokes–Poisson–Nernst–Planck equations

This work is concerned with the global existence of large solutions to the three‐dimensional dissipative fluid‐dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants such that if the initial data satisfies then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution.

Optimal control of deterministic and stochastic Eco-epidemic food adulteration model

In this paper, we develop a deterministic and stochastic mathematical model to study the effect of contaminated foods on human health by spreading media awareness. A predator–prey SI model with media-induced response function is considered. In the deterministic case, the model has three equilibria: trivial, disease-free, and endemic. While the endemic equilibria exist for low media impact provided R0>1. Here, the trivial equilibrium is always unstable. However, the disease-free equilibrium exhibits local and global asymptotically stable behavior provided that R0<1. The conditions for the local behavior of endemic equilibrium are also determined along with the existence of Hopf bifurcation. Further, the impact of external interference is studied for the stochastic model. The stochastic model has a unique global positive solution which is ultimately bounded. The stochastic model also demonstrates asymptotic behavior around disease-free and endemic equilibria. Additionally, optimal control theory has been employed to explore deterministic and stochastic optimal control systems in depth. Finally, the analytical results are validated using some numerical simulations.

Transmission of drug-resistant bacteria in a hospital-community model stratified by patient risk

A susceptible-infectious-susceptible (SIS) model for simulating healthcare-acquired infection spread within a hospital and associated community is proposed. The model accounts for the stratification of in-patients into two susceptibility-based risk groups. The model is formulated as a system of first-order ordinary differential equations (ODEs) with appropriate initial conditions. The mathematical analysis of this system is demonstrated. It is shown that the system has unique global solutions, which are bounded and non-negative. The basic reproduction number (mathscr {R}_0) for the considered model is derived. The existence and the stability of the stationary solutions are analysed. The disease-free stationary solution is always present and is globally asymptotically stable for mathscr {R}_0<1, while for mathscr {R}_0>1 it is unstable. The presence of an endemic stationary solution depends on the model parameters and when it exists, it is globally asymptotically stable. The endemic state encompasses both risk groups. The endemic state within only one group only is not possible. In addition, for mathscr {R}_0=1 a forward bifurcation takes place. Numerical simulations, based on the anonymised insurance data, are also presented to illustrate theoretical results.

Dynamical analysis on stochastic two-species models

In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results.

A hyperbolic-elliptic-parabolic PDE model of chemotactic E. coli colonies

This paper studies the hyperbolic-elliptic-parabolic system{ut=−∇⋅(u∇c)+g(u)nu−b(n)u1+θ,(x,t)∈Ω×R+,0=Δc+αu−βc,(x,t)∈Ω×R+,nt=Δn−γg(u)nu,(x,t)∈Ω×R+,∂c∂ν=∂n∂ν=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),n(x,0)=n0(x),x∈Ω in a bounded domain Ω⊂RN(N≥1) with smooth boundary, where g(s) and b(s) are nonnegative smooth functions, α,β,γ>0 and θ∈[0,∞). We prove that if θ>1 or θ=1 with lims→+∞⁡b(s)=b0>α, the system admits a unique global strong solution, while if θ∈[0,1), the system exhibits a finite-time explosion phenomenon in the sense of L∞ norm for some large ‖u0‖Lp(Ω) with p>N. Here the convexity of the domain Ω is not required.

Global strong solutions to the one-dimensional compressible Navier-Stokes-Allen-Cahn system with phase field variable dependent viscosity

In this paper, we study the global existence and large-time behavior of strong solutions to the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system with phase field variable dependent viscosity, which models the motion of a mixture of two viscous compressible fluids. The case when the pressure p(ρ)=ργ, the viscosity ν(ρ,χ)=χα, the interface thickness δ(ρ)=ρβ and the relaxation time function a(ρ,χ,χy)=χλ is considered, where ρ>0 and χ are the density and the phase field variable, respectively, and γ,α,β,λ∈R are parameters. Under some conditions on the parameters γ,α,β,λ and the initial data, it is shown that the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system admits a unique global strong nonvacuum solution, which tends to the constant equilibrium state as time goes to infinity. Moreover, the exponential time decay rate of the phase field variable χ is also obtained. Here the initial perturbation for the phase field variable χ should be small, but the initial perturbations for the density and velocity of the fluid can be arbitrarily large.

Spatiotemporal Dynamics and Bifurcation Analysis of a Generalized Two-Prey One-Predator System with Diffusion and Double Prey-Taxes

The presence of a predator can force the mediation of a coexistence state in three-species ordinary differential equation model, where two competing species are preyed on by a common predator. To understand how the addition of diffusion and prey-taxis affects predator-mediated coexistence in such an ecological system, we consider a general two-competing-prey and one-predator model with double prey-taxes under Neumann boundary conditions. We first show that there is a unique global classical solution to this model with ratio-dependent and nonratio-dependent predator functional responses. Then, we demonstrate the emergence of the so-called stationary patterns. Finally, in detail, we give some sufficient conditions for the existence, nonexistence, and stability of nonconstant positive steady states and time-periodic positive solutions. Surprisingly, we find that the combination of a repulsive prey-taxis and an attractive prey-taxis can also induce the emergence of pattern formations. The theoretical results imply that double prey-taxes play an extremely important part in biological control.

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

Motivated by [2,3], we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.

Dynamical behavior of a stochastic HIV model with logistic growth and Ornstein-Uhlenbeck process

In this paper, we investigate a stochastic human immunodeficiency virus (HIV) model with logistic growth and Ornstein-Uhlenbeck process, which is used to describe the pathogenesis and transmission dynamics of HIV in the population. We first validate that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient conditions for the existence of a stationary distribution of the system, which shows the coexistence of all CD4+ T cells and free viruses. Especially, under some mild conditions which are used to ensure the local asymptotic stability of the quasi-chronic infection equilibrium of the stochastic system, we obtain the specific expression of covariance matrix in the probability density around the quasi-chronic infection equilibrium of the stochastic system. In addition, for completeness, we also obtain sufficient criteria for elimination of all infected CD4+ T cells and free virus particles. Finally, several examples together with comprehensive numerical simulations are conducted to support our analytic results.

Dynamical behavior of a stochastic dengue model with Ornstein–Uhlenbeck process

We develop and study a stochastic dengue model with Ornstein–Uhlenbeck process, in which we assume that the transmission coefficients between vector and human satisfy the Ornstein–Uhlenbeck process. We first show that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient criteria for the existence of a stationary distribution of the system, which indicates the persistence of the disease. In particular, under some mild conditions which are applied to ensure the local asymptotic stability of the endemic equilibrium of the deterministic system, we obtain the specific form of covariance matrix in the probability density around the quasi-positive equilibrium of the stochastic system. In addition, we also establish sufficient criteria for wiping out of the disease. Finally, several numerical simulations are performed to illustrate our theoretical conclusions.

Existence of global solutions to the nonlocal Schrödinger equation on the line

AbstractWe address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation under the initial data with the small‐norm. The nonlocal NLS equation was first introduced by Ablowitz and Musslimani as a new nonlocal reduction of the well‐known Ablowitz–Kaup–Newell–Segur system. The main technical difficulty for proving its global well‐posedness on the line in is due to the fact that mass and energy conservation laws, being nonlocal, do not preserve any reasonable norm and may be negative. In this paper, we use the inverse scattering transform approach to prove the existence of global solutions in based on the representation of a Riemann–Hilbert (RH) problem associated with the Cauchy problem of the nonlocal NLS equation. A key of this approach is, by applying the Volttera integral operator and Cauchy integral operator, to establish a Lipschitz bijective map between the solution of the nonlocal NLS equation and reflection coefficients associated with the RH problem. By using the reconstruction formula and estimates on the solution of the time‐dependent RH problem, we further affirm the existence of a unique global solution to the Cauchy problem for the nonlocal NLS equation.

Dynamics analysis of a stochastic HIV model with non-cytolytic cure and Ornstein–Uhlenbeck process

In this paper, we introduce an HIV infection model with virus-to-cell infection, cell-to-cell infection and non-cytolytic cure. Two mean-reverting Ornstein–Uhlenbeck processes are also taken into account in the model. Firstly, it is proved that the stochastic model has a unique positive global solution. The model is found to have at least one stationary distribution by constructing suitable Lyapunov functions if the critical condition R0s>1. Then, the probability density function near the quasi-positive equilibrium is obtained by solving the corresponding Fokker–Planck equation. The spectral radius method is used to derive the virus extinction under a sufficient condition R0e<1. Finally, some numerical simulations are carried out.

The evolution Navier–Stokes equations in a cube under Navier boundary conditions: rarefaction and uniqueness of global solutions

We study the evolution Navier–Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations.

Global boundedness in a chemotaxis system with signal-dependent motility and indirect signal consumption

This paper deals with the following chemotaxis system ut=Δuγv,x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≤3), where the motility function γ∈C30,+∞ is positive on [0,∞). For all suitably regular initial data, then the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. The purpose of this work is to remove the smallness assumption on ‖v0‖L∞(Ω) in three dimension (Xu et al., 0000).

Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

Lévy impact on the transmission of worms in wireless sensor network: Stochastic analysis

This manuscript addresses the security challenges wireless sensor networks (WSNs) face due to their operational limitations. The primary challenge stems from the infiltration of worms into the network, where one infected node could uncontrollably propagate the malware to neighboring node(s). First, we proposes a stochastic system based on Lévy noise to explain the spread of worms in WSNs. Then, we establish a unique positive global solution for the proposed model. We also examine the presence and potential extinction of worms within the networks. The results reveal that random environmental perturbations can confine the spread of worms and that the deterministic model tends to overestimate the worms’ spreading capacity. Using different parameter sets, the study obtains approximate solutions to validate these analytical findings and demonstrate the effectiveness of the suggested SEIRS system. The findings of the work reveal that the proposed model surpasses existing models in mitigating worm transmission in WSNs. Our inference suggests that the transmission dynamics of the system are influenced by both white noise and Lévy noise.

Global well‐posedness for the one‐phase Muskat problem

AbstractThe free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.