Unique Global Classical Solution Research Articles

Study on a growth-expansion model with chemotaxis in nutrient-replete environments

We consider the no-flux initial-boundary value problem for the growth-expansion model with chemotaxis in nutrient-replete environments in smoothly bounded domains [Formula: see text]. It is shown that if [Formula: see text] or if [Formula: see text] under some structural assumptions on parameter functions therein, for any suitably regular initial data the problem admits a unique global bounded classical solution. For any dimensions [Formula: see text], we also prove that the problem has a unique global classical solution which is bounded under a small assumption on the initial data. Moreover, we obtain that these solutions stabilize to a uniquely determined spatially uniform equilibrium. We also provide exponential rates of convergence of solutions in a special case.

Global boundedness and asymptotic behavior in a double haptotaxis model for oncolytic virotherapy

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy{ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v−κv,wt=DwΔw−ξw∇⋅(w∇v)−w+ρuz,zt=DzΔz−z−ρuz+βw, in a smooth bounded domain Ω⊂R2, with parameters Dw>0, Dz>0, ξw≥0, ρ≥0, κ>0 and β≥0. This system describes the interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is bounded in (L∞(Ω))4. Moreover, it is shown that this bounded solution can approach a spatially constant equilibrium in the large time limit under the additional assumption that 0≤β<1.

Boundedness in a chemotaxis-May–Nowak model with exposed state

This paper is concerned with the Neumann initial–boundary value problem of the chemotaxis-May–Nowak model with exposed state for virus dynamics. It is proved that the problem admits a unique global classical solution which is uniformly bounded for all sufficiently smooth initial data in smoothly bounded domains Ω⊂Rn, n≤3.

On small-data solution of the chemotaxis–SIS epidemic system with bilinear incidence rate

This paper deals with the initial–boundary value problem for chemotaxis susceptible-infected-susceptible epidemic system with bilinear incidence rate ut=d1Δu+χ∇⋅(u∇v)−β(x)uv+γ(x)v,x∈Ω,t>0,vt=d2Δv+β(x)uv−γ(x)v,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0, χ∈R, 0<β∈C1(Ω¯) and 0<γ∈C1(Ω¯). It is proved that if (u(x,0),v(x,0)) in L∞(Ω)×L1(Ω) is suitably small and ‖v(x,0)‖L∞(Ω)+‖∇v(x,0)‖Lq(Ω)≤K for each K>0 and q>n, then the problem possesses a unique global classical solution which is bounded in Ω×(0,∞). Moreover, we prove that the solution exponentially stabilizes to the disease-free equilibrium N|Ω|,0 with N=∫Ω(u(x,0)+v(x,0)) in L∞(Ω) as t→∞.

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

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 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).

Global existence and time-decay rates of classical solutions to the generalized incompressible Oldroyd-B model in [formula omitted

In this paper, we investigate the initial value problem for the three dimensional generalized incompressible Oldroyd-B model. Based on the energy estimates and the tricky analytical skills, we prove that the problem always exists a unique global classical solution without any smallness restriction on the higher order spatial derivatives of initial data. Moreover, the optimal time-decay rates of solutions itself and the higher-order spatial derivatives of solutions are obtained without linear decay analysis. The result implies that the negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rate.

Global classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions

In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.

Classical solutions to a model of two-phase incompressible flows with variable density

We study the global well-posedness of three-dimensions Allen-Cahn-Navier-Stokes equations, a diffuse-interface model for two-phase incompressible flows with different densities. We first prove the local-in-time existence of classical solutions with finite initial energy. The key point is to carefully design an energy norm. The derivative f′(ϕ) of the physical relevant energy density f(ϕ) produces a damping effect near equilibrium ϕ=±1. We thereby establish the unique global classical solution near equilibrium under small size of initial energy.

Global existence and stabilization in a two-dimensional chemotaxis-Navier-Stokes system with consumption and production of chemosignals

We are concerned with the initial boundary value problems of chemotaxis-Navier-Stokes systems involving double chemosignals: one is an attractant consumed by the cells themselves and the other is an attractant or a repellent produced by the cells themselves. This work reveals that for any properly regular initial data, if the initial total cells mass is less than some threshold, then the corresponding attraction-atraction Navier-Stokes system admits a unique global classical solution, which, under extra two smallness assumptions on the initial data, is globally bounded and converges at exponential rate to the spatial equilibrium in the large time limit. As byproducts, the global classical solvability and stabilization of the Keller-Segel Navier-Stokes system and the uniform boundedness and large time behavior of the global classical solution of the associated attraction-repulsion Navier-Stokes system are simultaneously achieved.

Dynamics analysis on a chronic epidemic model with diffusion

In this paper, we propose a reaction-diffusion model of chronic infectious diseases with a special nonlinear incidence rate describing the mechanism of some chronic infections diseases. We first prove that the model has a unique nonnegative global classical solution, and admits a connected global attractor. Further, the threshold dynamics such as the uniform persistence of the model solution and the global stability of the steady-state solution are studied. The theory is verified by numerical simulation. Our results show that chronic infectious diseases have a strong ability to survive, and it is difficult to eradicate chronic infectious diseases completely.

Boundedness and asymptotic stability in a chemotaxis model with indirect signal production and logistic source

This article concerns the chemotaxis-growth system with indirect signal production $$\displaylines{ u_t=\Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega,\; t&gt;0,\cr 0=\Delta v-v+w,\quad x\in \Omega,\; t&gt;0,\cr w_t=-\delta w+u,\quad x\in\Omega,\; t&gt;0, }$$ on a smooth bounded domain \(\Omega\subset \mathbb{R}^n\) (\(n\geq1\) with homogeneous Neumann boundary condition, where the parameters \(\mu, \delta&gt;0\). It is proved that if \(n\leq 2\) and \(\mu&gt;0\), for all suitably regular initial data, this model possesses a unique global classical solution which is uniformly-in-time bounded. While in the case \(n\geq 3\), we show that if \(\mu\) is sufficiently large, this system possesses a global bounded solution. Furthermore, the large time behavior and rates of convergence have also been considered under some explicit conditions.

Global classical solutions to the compressible micropolar viscous fluids with large oscillations and vacuum

In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows. We prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large oscillations and the initial density may allowed to contain the interior and far field vacuum states. Furthermore, the large time behavior of the solution is obtained as well.

On existence of global classical solutions to the 3D compressible MHD equations with vacuum

In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on mathbb{R}^{3}times [0, T] with any Tin (0, infty ), provided that the initial magnetic field in the L^{3}-norm and the initial density are suitably small. Note that the first result is obtained under the condition of rho _{0}in L^{gamma }cap W^{2, q} with qin (3, 6) and gamma in (1, 6). It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., rho _{0} notin L^{1}). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved.

Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

&lt;p style='text-indent:20px;'&gt;This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.&lt;/p&gt;

Dampening effects on global boundedness and asymptotic behavior in an oncolytic virotherapy model

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy in two-dimensional domains, accounting for interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is uniformly bounded under suitable assumptions. Moreover, it is shown that a bounded solution can approach spatially constant equilibrium in the large time limit.

Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi coordinates

We consider a characteristic initial value problem, with initial data given on a future null cone, for the Einstein (massless) scalar field system with a positive cosmological constant, in Bondi coordinates. We prove that, for small data, this system has a unique global classical solution which is causally geodesically complete to the future and decays polynomially in radius and exponentially in Bondi time, approaching the de Sitter solution.

Dynamics in an attraction–repulsion Navier–Stokes system with signal-dependent motility and sensitivity

This paper is concerned with an attraction–repulsion Navier–Stokes system with signal-dependent motility and sensitivity in a two-dimensional smooth bounded domain under zero Neumann boundary conditions for n, c, v and the homogeneous Dirichlet boundary condition for u. This system describes the evolution of cells that react on two different chemical signals in a liquid surrounding environment and models a density-suppressed motility in the process of stripe pattern formation through the self-trapping mechanism. The major difficulty in analysis comes from the possible degeneracy of diffusion as c and v tend to infinite. Based on a new weighted energy method, it is proved that under appropriate assumptions on parameter functions, this system possesses a unique global classical solution, which is uniformly-in-time bounded. Moreover, by means of energy functionals, it is shown that the global bounded solution of the system exponentially converges to the constant steady state.

Global boundedness and large time behavior of solutions to a chemotaxis–consumption system with signal-dependent motility

This paper deals with the following chemotaxis–consumption system with signal-dependent motility $$\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta (\gamma (v)u),\,\,\, &{}x\in \Omega ,\,\,\, t>0,\\ v_{t}=\Delta v-uv, \,\,\,&{}x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}$$ under no-flux boundary conditions in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^{n}$$ . $$\gamma (s)$$ is the motility function. For the case of positive motility function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, it is asserted that the solution to the system exponentially converges to constant equilibria in the large time. Finally, if the motility function is zero at some point, we obtain the existence of the weak solution.