Asymptotic Behavior Of Classical Solutions Research Articles

Global existence, boundedness and asymptotic behavior of classical solutions to a fully parabolic two-species chemotaxis-competition model with singular sensitivity

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics(0.1){ut=Δu−χ1∇⋅(uw∇w)+μ1u(1−u−a1v),t>0,x∈Ω,vt=Δv−χ2∇⋅(vw∇w)+μ2v(1−v−a2u),t>0,x∈Ω,wt=Δw−w+u+v,t>0,x∈Ω,∂u∂ν=∂v∂ν=∂w∂ν=0,t>0,x∈∂Ω,u(0,x)=u0(x),v(0,x)=v0(x),w(0,x)=w0(x),x∈Ω, where Ω⊂RN(N≥2) is a bounded smooth convex domain, and the parameters χ1,χ2,μ1,μ2,a1 and a2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ1,χ2∈(0,12)forN=2,3 or χ1,χ2∈(0,N−2N−1)forN≥4. Moreover, if min⁡{μ1,μ2}>max⁡{χ1,χ2}24, we obtain the uniformly lower bound for w. Finally, when χ1,χ2 are suitably small, it is proved that if 0<a1,a2<1, then the solution (u,v,w) converges to (1−a11−a1a2,1−a21−a1a2,2−a1−a21−a1a2) in L∞ norm as t→∞; if 0<a2<1≤a1, then the solution (u,v,w) converges to (0,1,1) in L∞ norm as t→∞.

Small solutions of the Einstein–Boltzmann-scalar field system with Bianchi symmetry

We show that small homogeneous solutions to the Einstein–Boltzmann-scalar field system exist globally toward the future and tend to the de Sitter solution in a suitable sense. More specifically, we assume that the spacetime is of Bianchi type I–VIII, that the matter is described by Israel particles and that there exists a scalar field with a potential which has a positive lower bound. This represents a generalization of the work [H. Lee and E. Nungesser, Classical Quantum Gravity 35, 025001 (2018)], where a cosmological constant was considered, and a generalization of [H. Lee and J. Lee, J. Math. Phys. 63, 031502 (2022)], where a spatially flat FLRW spacetime was considered. We obtain the global existence and asymptotic behavior of classical solutions to the Einstein–Boltzmann-scalar field system for small initial data.

Estimate of largest velocities for Vlasov–Poisson plasma with radiation damping

In this article, we study the asymptotic behavior of classical solutions to three‐dimensional Vlasov–Poisson system with radiation damping arising in plasma physics. For smooth solutions with compact support in phase space, the growth estimate of the largest velocity of plasma particles is improved to . As a consequence, some better estimates of the charge density and electrostatic field are obtained.

On a Relativistic BGK Model for Polyatomic Gases Near Equilibrium

Recently, a novel relativistic polyatomic BGK model was suggested by Pennisi and Ruggeri [J. Phys. Conf. Ser., 1035 (2018), 012005] to overcome drawbacks of the Anderson--Witting model and the Marle model. In this paper, we prove the unique existence and asymptotic behavior of classical solutions to the relativistic polyatomic BGK model when the initial data is sufficiently close to a global equilibrium.

Asymptotic behavior of classical solutions of a three-dimensional Keller–Segel–Navier–Stokes system modeling coral fertilization

We are concerned with the Keller--Segel--Navier--Stokes system \begin{equation*} \left\{ \begin{array}{ll} \rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho \mathcal{S}(x,\rho,c)\nabla c)-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\ m_t+u\cdot\nabla m=\Delta m-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\ c_t+u\cdot\nabla c=\Delta c-c+m, & \!\! (x,t)\in \Omega\times (0,T), \\ u_t+ (u\cdot \nabla) u=\Delta u-\nabla P+(\rho+m)\nabla\phi,\quad \nabla\cdot u=0, &\!\! (x,t)\in \Omega\times (0,T) \end{array}\right. \end{equation*} subject to the boundary condition $(\nabla\rho-\rho \mathcal{S}(x,\rho,c)\nabla c)\cdot \nu\!\!=\!\nabla m\cdot \nu=\nabla c\cdot \nu=0, u=0$ in a bounded smooth domain $\Omega\subset\mathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $\mathcal{S}\in C^2(\overline\Omega\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,\rho,c)|\leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.

Explicit decay rates for a generalized Boussinesq–Burgers system

This paper is concerned with the long-time asymptotic behavior of classical solutions to the Cauchy problem for the generalized Boussinesq–Burgers system: ut+(uw)x=εuxx,x∈R,t>0,wt+uγ+w22x=μwxx+δwxxt,x∈R,t>0,(u,w)(x,0)=(u0,w0)(x),x∈R,where γ≥2, ε, μ and δ are positive constants. By utilizing time-weighted energy methods, we identify the explicit decay rates of classical solutions to the Cauchy problem under mild conditions on the initial data. This generalizes the previous result obtained in Zhu and Liu (2016) by extending the exponent γ from a single value to the half real line.

Global existence and asymptotic behavior of classical solutions to a fractional logistic Keller–Segel system

In this paper we consider a fractional parabolic–elliptic Keller–Segel system with a logistic source on RN. First, we establish the regularity of weak solutions of the fractional parabolic equation, using blow-up arguments combined with Liouville-type theorems. Next, by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions. Moreover, the global existence and boundedness of classical solutions for given initial data are obtained under some conditions. Finally, we show the asymptotic behavior of the global solutions with strictly positive initial data.

Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier–Stokes Equations with Vacuum

This paper concerns the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For strong and classical solutions, some a priori decay with rates (in large time) for both the pressure and the spatial gradient of the velocity field are obtained provided that the initial total energy is suitably {small.} Moreover, by using these key decay rates and some analysis on the expansion rates of the essential support of the density, we establish the global existence and uniqueness of classical solutions (which may be of possibly large oscillations) in two spatial dimensions, provided the smooth initial data are of small total energy. In addition, the initial density can even have compact support. This, in particular, yields the global regularity and uniqueness of the re-normalized weak solutions of Lions-Feireisl to the two-dimensional compressible barotropic flows for all adiabatic number $\gamma>1$ provided that the initial total energy is small.

Global Existence and Asymptotic Behavior of Classical Solutions for a 3D Two-Species Keller-Segel-Stokes System with Competitive Kinetics

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n_1)_t + u\cdot\nabla n_1 =\Delta n_1 - \chi_1\nabla\cdot(n_1\nabla c)+ \mu_1n_1(1- n_1 - a_1n_2)$, $(n_2)_t + u\cdot\nabla n_2 =\Delta n_2 - \chi_2\nabla\cdot(n_2\nabla c) + \mu_2n_2(1- a_2n_1 - n_2), c_t + u\cdot\nabla c =\Delta c - c + \alpha n_1 +\beta n_2$, $u_t= \Delta u + \nabla P+ (\gamma n_1 + \delta n_2)\nabla\phi$, $ \nabla\cdot u = 0$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+\alpha n_1+\beta n_2$ is replaced with $-(\alpha n_1+\beta n_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $\mu_1,\mu_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^\infty$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $\mu_1,\mu_2$.

Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate

We consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal BGK model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic Maxwellian. We observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part, leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter θ reaches zero. Accordingly, we employ two different sets of micro–macro system to derive the full coercivity and close the energy estimate.

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics urn:x-wiley:mma:media:mma4807:mma4807-math-0001 under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small.

Global existence and asymptotic behavior of classical solutions to a parabolic–elliptic chemotaxis system with logistic source on [formula omitted

In the current paper, we consider the following parabolic–elliptic semilinear Keller–Segel model on RN,{ut=∇⋅(∇u−χu∇v)+au−bu2,x∈RN,t>0,0=(Δ−I)v+u,x∈RN,t>0, where χ>0,a≥0,b>0 are constant real numbers and N is a positive integer. We first prove the local existence and uniqueness of classical solutions (u(x,t;u0),v(x,t;u0)) with u(x,0;u0)=u0(x) for various initial functions u0(x). Next, under some conditions on the constants a,b,χ and the dimension N, we prove the global existence and boundedness of classical solutions (u(x,t;u0),v(x,t;u0)) for given initial functions u0(x). Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants a,b,χ and the dimension N, we show that for every strictly positive initial function u0(⋅),limt→∞⁡supx∈RN⁡[|u(x,t;u0)−ab|+|v(x,t;u0)−ab|]=0, and that for every nonnegative initial function u0(⋅) with non-empty and compact support supp(u0), there are 0<clow⁎(u0)≤cup⁎(u0)<∞ such thatlimt→∞⁡sup|x|≤ct⁡[|u(x,t;u0)−ab|+|v(x,t;u0)−ab|]=0∀0<c<clow⁎(u0) andlimt→∞⁡sup|x|≥ct⁡[u(x,t;u0)+v(x,t;u0)]=0∀c>cup⁎(u0).

Global asymptotic behavior of large solutions for a class of semilinear elliptic problems

By using Karamata regular variation theory and upper and lower solution method, we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlinear convection terms. In our study, the weight and nonlinearity are controlled by some regularly varying functions or rapid functions, which is very different from the conditions of previous contexts. Our results largely extend the previous works, and prove that the nonlinear convection terms do not affect the global asymptotic behavior of classical solutions when the index of the convection terms change in a certain range.

Global existence and asymptotic behavior for the full compressible Euler equations with damping in $\mathbb R^3$

We are concerned with the global existence and asymptotic behavior of classical solutions to the Cauchy problem for the full compressible Euler equations with damping in $\mathbb R^3$. We prove the global existence of the classical solutions by the delica

The 3D non-isentropic compressible Euler equations with damping in a bounded domain

The authors investigate the global existence and asymptotic behavior of classical solutions to the 3D non-isentropic compressible Euler equations with damping on a bounded domain with slip boundary condition. The global existence and uniqueness of classical solutions are obtained when the initial data are near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

Large Time Asymptotic Behavior of Classical Solutions to the 3D Compressible Nematic Liquid Crystal Flows with Vacuum

This paper concerns the large time asymptotic decay with rates of the global classical solutions in the three spatial dimensions with vacuum as far field density. we prove that the L2$L^{2}$-norm of both the pressure, the gradient of the velocity and the gradient of orientation decay in time with a rate tź12$t^{-\frac{1}{2}}$, and the gradient of the vorticity and the effective viscous flux decay faster than themselves. When d$d$ is a constant vector, the large time decay rates (1.12) are the same as Li and Xin (Global well-posedness and decay asymptotic behavior of classical solution to the compressible Navier-Stokes equations with vacuum, 2003).

Global existence and decay rate of the Boussinesq–Burgers system with large initial data

In this paper, we study the global existence and the long-time asymptotic behavior of classical solutions to the Cauchy problem for the Boussinesq–Burgers system with large initial data. More precisely, we first show that the classical solutions exist globally in time with large initial data by using the Lp (p>2) estimates other than the conventional L2 estimate. Then we prove that the global classical solutions converge to constant equilibrium states with an algebraic decay rate as time approaches infinity.

Asymptotic behavior of classical solutions to the compressible Navier–Stokes–Poisson equations in three and higher dimensions

In this paper, we consider the initial value problem for the compressible Navier–Stokes–Poisson equations in three and higher dimensions. Firstly, global existence and new decay estimate of classical solutions are established. The proof is mainly based on the decay properties of solution operator to the compressible Navier–Stokes–Poisson equations, which may be derived from the pointwise estimate of the solution operator to a linear wave equation that is corresponding to the density function, while the pointwise estimate of solution operator to the linear wave equation may be established by an energy method in the Fourier space. The method is different from those in previous works on decay estimate of solutions to compressible Navier–Stokes–Poisson equations. Finally, asymptotic profile of the solution to the corresponding linear problem is also investigated.

Global Existence and Asymptotic Behavior for the 3D Compressible Non–isentropic Euler Equations with Damping

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

Global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system

In this article, we investigate the global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system. Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C1 solutions with the bounded L1 ∩ L ∞ norm of the boundary data as well as their derivatives. Based on the existence result, we can prove that when t tends to in nity, the solutions approach a combination of piece-wised C1 traveling wave solutions. As the important example, we apply the results to the chaplygin gas system. Mathematics Subject Classi cation (2000): 35B40; 35L50; 35Q72.