Large Time Behavior Of Solutions Research Articles

Global bounded weak solutions and asymptotic behavior to a chemotaxis-Stokes model with non-Newtonian filtration slow diffusion

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p>2){nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−χ∇⋅(n∇c),ct+u⋅∇c−Δc=−cn,ut+∇π=Δu+n∇φ,divu=0 in a bounded domain Ω of R3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value (p≥2) which ensures that the solution is global bounded. In particular, the closer the value of p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist wheneverp>p⁎(≈2.012). It improved the result of [21,22], in which, the authors established the global bounded solutions for p>2311. Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state (n‾0,0,0). Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of L∞-norm, not only in Lp-norm or weak-* topology.

Global Existence and Large Time Behavior of Solutions to 3D MHD System Near Equilibrium

In this paper, we consider the stability problem on perturbation near a physically steady state solution of the 3D generalized incompressible magnetohydrodynamic system in Lei-Lin space. The global stability and analytic estimates for small perturbation are established by the semigroup method in the critical space $$\chi ^{1-2\alpha }(\mathbb {R}^3)$$ with $$\frac{1}{2}\le \alpha \le 1$$ , where linear terms from perturbation incur much difficulty. By introducing a diagonalization process we successfully eliminate the linear terms. Then, by virtue of the analytic estimates for a solution, the temporal decay rate $$(1+t)^{-(\frac{5}{4\alpha }-1)}$$ of the global solution is obtained.

Large time behaviour for the heat equation on [formula omitted], moments and decay rates

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh Z on ℓp spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and ℓp decay results for the fundamental solution. We use that estimates to get rates on the ℓp decay and large time behaviour of solutions. For the ℓ2 case, we get optimal decay by use of Fourier techniques.

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.

The 3D Vlasov–Poisson–Landau System Near 1D Local Maxwellians

We consider the Cauchy problem on the Vlasov–Poisson–Landau system with the Coulomb interaction in the three dimensional space domain $${\mathbb {R}}\times {\mathbb {T}^2}$$ . Although there have been extensive studies on global existence and large time behavior of solutions near global Maxwellians either in $$\mathbb {T}^3$$ or $$\mathbb {R}^3$$ , it is unknown whether solutions can be constructed around some non-trivial asymptotic profiles. In this paper, we obtain the global solutions near a spatially one-dimensional local Maxwellian connecting two distinct global Maxwellians at $$x_1=\pm \infty $$ , where the fluid components of the local Maxwellian are the smooth approximate rarefaction wave of the corresponding full compressible Euler system in $$x_1\in \mathbb {R}$$ . We also prove the large time asymptotics of global solutions towards such planar rarefaction waves, and establish the propagation of the high-order moments and regularity of solutions with large amplitude. As a byproduct, all these results can be carried over to the pure Landau equation in the same setting.

Large time behavior of solutions of Trudinger's equation

We study the large time behavior of solutions v:Ω×(0,∞)→R of the PDE ∂t(|v|p−2v)=Δpv. We show that e(λp/(p−1))tv(x,t) converges to an extremal of a Poincaré inequality on Ω with optimal constant λp, as t→∞. We also prove that the large time values of solutions approximate the extremals of a corresponding “dual” Poincaré inequality on Ω. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.

Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions

This work is concerned with the large time behavior of solutions to the barotropic compressible Navier-Stokes equations in Rd(d≥2). A pure energy argument (independent of spectral analysis) has been developed in more general Lp framework, which enables us not only to obtain the optimal time-decay rates but also to remove the smallness of low frequencies of initial data that has been required in previous studies on this problem. The crucial part of the proof lies in the evolution of negative Besov norms at low frequencies, which mainly depends on non standardLp product estimates and some elaborate use of Sobolev embeddings and interpolations. Our results could also hold true in case of large highly oscillating initial velocities in comparison with classical efforts.

A Class of Sixth Order Viscous Cahn-Hilliard Equation with Willmore Regularization in ℝ3

The main purpose of this paper is to study the Cauchy problem of sixth order viscous Cahn–Hilliard equation with Willmore regularization. Because of the existence of the nonlinear Willmore regularization and complex structures, it is difficult to obtain the suitable a priori estimates in order to prove the well-posedness results, and the large time behavior of solutions cannot be shown using the usual Fourier splitting method. In order to overcome the above two difficulties, we borrow a fourth-order linear term and a second-order linear term from the related term, rewrite the equation in a new form, and introduce the negative Sobolev norm estimates. Subsequently, we investigate the local well-posedness, global well-posedness, and decay rate of strong solutions for the Cauchy problem of such an equation in R3, respectively.

Global Wellposedness and Large Time Behavior of Solutions to the Hall-Magnetohydrodynamics Equations

We prove the global solutions to the incompressible Hall-magnetohydrodynamics (Hall-MHD) equations with small or some large initial data in \mathbb R^n (n \geq 3) . Especially, Fujita–Kato type initial data as the incompressible Navier–Stokes equations are allowed. We also study the large time behavior of the solutions and obtain an optimal decay rate in the general Besov spaces. Different from all the previous wellposedness results in [6, 7, 21], the initial energy we considered here may not be finite.

Remarks on exponential stability for a coupled system of elasticity and thermoelasticity with second sound

We study the large time behavior of solutions to a linear transmission problem in one space dimension. The problem at hand models a thermoelastic material with second sound confined by a purely elastic one. We shall characterize all equilibrium states of the considered system and prove that every solution approaches one designated equilibrium state with an exponential rate as time goes to infinity. Hereto, we apply methods from the theory of strongly continuous semigroups. In particular, we obtain uniform resolvent bounds for the underlying generator. This removes the largeness assumption of elastic wave speeds imposed in [Y.P. Meng and Y.G. Wang, Anal. Appl. (Singap.) 13 (2015)] for having an exponential energy decay rate when the problem only has the trivial equilibrium. In an appendix we provide a similar exponential stability result for the case where heat conduction is modeled using Fourier's law.

Large time behaviour of solutions to the 3D Navier–Stokes equations with damping

In this paper, we consider large time behaviour of solutions to the 3D Navier–Stokes equations with damping term $$|u|^{\beta -1}u$$ ( $$\beta \ge 1$$ ). For $$\beta >\frac{7}{3}$$ , we derive decay rates of the $$L^2$$ -norm of the solutions. By using Fourier splitting method, we also prove that the solutions are asymptotically equivalent to the solutions of the classic 3D Navier–Stokes equations.

Diffusion wave phenomena and Lp decay estimates of solutions of compressible viscoelastic system

We consider the system of equations describing motion of compressible viscoelastic fluids in a whole space. We investigate the large time behavior of solutions around a motionless state, and obtain the Lp decay estimates of solutions for 1<p≤∞, provided that the initial data is sufficiently close to the motionless state. In addition, we clarify the diffusion wave phenomena caused by the sound wave and the elastic shear wave.

Large time behavior of solutions to space-time monopole equations in 1 + 1 dimensions

We study the large time behavior of solutions to the space-time monopole equations in [Formula: see text] dimensions. We establish that the solutions will tend to traveling wave solution when time tends to infinity.

Asymptotic Stability of a Viscous Contact Wave for the One-Dimensional Compressible Navier-Stokes Equations for a Reacting Mixture

We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2-energy methods.

Global solvability and stabilization to a cancer invasion model with remodelling of ECM * * This work is supported by NSFC (11871230) and Guangdong Basic and Applied Basic Research Foundation (2020B1515310013).

In this paper, we deal with the Chaplain–Lolas’s model of cancer invasion with tissue remodelling We consider this problem in a bounded domain (N = 2, 3) with zero-flux boundary conditions. We first establish the global existence and uniform boundedness of solutions. Subsequently, we also consider the large time behaviour of solutions, and show that the global classical solution (u, v, w) strongly converges to the semi-trivial steady state in the large time limit if δ > η; and strongly converges to if δ < η. Unfortunately, for the case δ = η, we only prove that (v, w) → (1, 0), and it is hard to obtain the large time limit of u due to lack of uniform boundedness of . It is worth noting that the large time behaviour of solutions for the chemotaxis–haptotaxis model with tissue remodelling has never been touched before, this paper is the first attempt. At last, taking advantage of the large time behaviour of solutions, we also establish the uniform boundedness of solutions in the classical sense.

Boundedness and large time behavior of solutions to a prey-taxis system accounting in liquid surrounding

In this paper, we are concerned with the following prey-taxis system with liquid surrounding describing by the incompressible Navier–Stokes equations nt+u⋅∇n=Δn−∇⋅(χn∇c)+γnF(c)−θn−αn2,x∈Ω,t>0,ct+u⋅∇c=DΔc−nF(c)+f(c),x∈Ω,t>0,ut+u⋅∇u=Δu+∇P+n∇ϕ,∇⋅u=0,x∈Ω,t>0,∂n∂ν=∂c∂ν=u=0,x∈∂Ω,t>0,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),x∈Ω,where Ω⊂R2 is a bounded domain with smooth boundary. Using the Lp-energy estimate, we obtain the global existence of solutions with uniform-in-time bound. Moreover, by constructing some Lyapunov functionals, we also establish the large time behavior of solutions.

Critical Fujita Exponent for the Porous Medium Equation in ${\mathbb R}^{{N}}$ with hole

We study the large time behavior of solutions to the Neumann Problem of the porous medium equation. It is shown that the critical Fujita exponent is determined not only by the spatial dimension and the nonlinear exponent, but also by the coefficient $k$ of the first-order term.

The large-time behavior of solutions in the critical Lp framework for compressible viscous and heat-conductive gas flows

The $L^{p}$ theory for non-isentropic Navier-Stokes equations governing compressible viscous and heat-conductive gases is not yet proved completely so far, because the critical regularity cannot control all non linear coupling terms. In this paper, we pose an additional regularity assumption of low frequencies in $\mathbb{R}^d(d\geq 3)$, and then the sharp time-weighted inequality can be established, which leads to the time-decay estimates of global strong solutions in the $L^{p}$ critical Besov spaces. Precisely, we show that if the initial data belong to some Besov space $\dot{B}^{-s_{1}}_{2,\infty}$ with $s_{1}\in (1-\frac{d}{2}, s_0](s_0\triangleq \frac{2d}{p}-\frac{d}{2})$, then the $L^{p}$ norm of the critical global solutions admits the time decay $t^{-\frac{s_{1}}{2}-\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}$ (in particular, $t^{-\frac{d}{2p}}$ if $s_1=s_0$), which coincides with that of heat kernel in the $L^p$ framework. In comparison with \cite{DX2}, the low-frequency regularity $s_1$ can be improved to be \textit{the whole range}.

On a comparison principle for Trudinger’s equation

Abstract We study the comparison principle for non-negative solutions of the equation ∂ ⁡ ( | v | p - 2 ⁢ v ) ∂ ⁡ t = div ⁡ ( | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v ) , 1 &lt; p &lt; ∞ . \frac{\partial(|v|^{p-2}v)}{\partial t}=\operatorname{div}(|\nabla v|^{p-2}% \nabla v),\quad 1&lt;p&lt;\infty. This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions.

Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all $ t&gt;0 $ step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the $ L^p $ decay estimate for the solution $ u(\cdot,t) $ and all its derivatives for generalized Navier-Stokes equations with $ \max\{1,\frac{3+q}6\}&lt;\alpha\leq\frac12+\min\{\frac3q-\frac3p,\frac3{2p}\} $.