Time Behavior Of Solutions Research Articles

Behavior in time of solutions to a degenerate chemotaxis system with flux limitation

We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system ut=∇⋅(u∇uu2+|∇u|2)−χkf∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0under no flux boundary conditions in a ball B=Ω⊂RN and initial condition u(x,0)=u0(x)>0,χ>0,α>0,kf>0 and μ=1|Ω|∫Ωu0dx. Under suitable conditions on α and u0 it is shown that the solution blows up in L∞-norm at a finite time Tmax and for some p>1 it blows up also in Lp-norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates.

Dynamic behavior of reaction–diffusion equation with Holling II predation

In this paper, we consider the positive solutions for some reaction–diffusion equations with Holling II predation. The main aim is to analyze the long time behavior of solutions, both the diffusion and non-diffusion problems are investigated. Our results also exhibit different effects between Dirichlet and Neumann boundary conditions.

Behavior in time of solutions of a Keller–Segel system with flux limitation and source term

In this paper we consider radially symmetric solutions of the following parabolic–elliptic cross-diffusion system ut=Δu-∇·(uf(|∇v|2)∇v)+g(u),0=Δv-m(t)+u,∫Ωvdx=0,u(x,0)=u0(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\Delta u - \ abla \\cdot (u f(|\ abla v|^2 )\ abla v) + g(u), &{} \\\\ 0= \\Delta v -m(t)+ u, \\quad \\int _{\\Omega }v \\,dx=0, &{} \\\\ u(x,0)= u_0(x), &{} \\end{array}\\right. } \\end{aligned}$$\\end{document}in Omega times (0,infty ), with Omega a ball in {mathbb {R}}^N, Nge 3, under homogeneous Neumann boundary conditions, where g(u)= lambda u - mu u^k, lambda>0, mu >0, and k >1, f(|nabla v|^2 )= k_f(1+ |nabla v|^2)^{-alpha }, alpha >0, which describes gradient-dependent limitation of cross diffusion fluxes. The function m(t) is the time dependent spatial mean of u(x, t) i.e. m(t):= frac{1}{|Omega |} int _{Omega } u(x,t) ,dx. Under smallness conditions on alpha and k, we prove that the solution u(x, t) blows up in L^{infty }-norm at finite time T_{max} and for some p>1 it blows up also in L^p-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on alpha or k, we prove that the solution is global and bounded in time.

Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

AbstractIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for $$J_{\mu }(u_{0})<0$$ J μ ( u 0 ) < 0 and decay rate of $$H^{1}_{0}$$ H 0 1 and $$L^{2}$$ L 2 -norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

Modified Scattering for the One-Dimensional Schrödinger Equation with a Subcritical Dissipative Nonlinearity

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity $$\lambda |u|^\alpha u$$ , where $$0<\alpha <2$$ , and $$\lambda $$ is a complex constant satisfying $$\text {Im} \lambda >\frac{\alpha |\text {Re} \lambda |}{2\sqrt{ \alpha +1}}$$ . For arbitrary large initial data, we present the uniform time decay estimates when $$4/3\le \alpha <2$$ , and the large time asymptotics of the solution when $$\frac{7+\sqrt{145}}{12}<\alpha <2$$ . The proof is based on the vector fields method and a semiclassical analysis method.

Local zero-stability of rough evolution equations

We analyze the long time behavior of solutions to rough parabolic equations. More precisely, we show local exponential stability for the mild solution driven by a fractional Brownian motion with Hurst parameter [Formula: see text].

Absence of singularities in solutions for the compressible Euler equations with source terms in

The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in $\mathbb {R}^d$ , $d\geq 1$ , which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the $L^2$ -norm of $\nabla u$ is exponential decay.

Stability and large-time behavior of the inviscid Boussinesq system for the micropolar fluid with damping

In this paper, we consider the three-dimensional inviscid Boussinesq system for the micropolar fluid in porous media. We proved the global well-posedness and large time behavior of solutions in the whole space R3. Precisely, when the H3-norm of initial data is small, but the higher-order derivatives can be arbitrarily large, the system is globally well-posed by the pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the Lp − L2 (1 ≤ p ≤ 2) type of the optimal time decay rates are obtained without any smallness assumption on the Lp norm of the initial data. Our results mathematically explain the stability of the system in an unbounded domain.

Global existence and blow-up for semilinear parabolic equation with critical exponent in R N

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].

$m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.

Asymptotic behavior in time of solutions to complex-valued nonlinear Klein-Gordon equation in one space dimension

We consider the long time behavior of solutions to the initial value problem for the ``complex-valued'' cubic nonlinear Klein-Gordon equation (NLKG) in one space dimension. In [12], Sunagawa derived the $L^{\infty}$ decay estimate of solutions to (NLKG). In this note, we obtain the large time asymptotic profile of solutions to (NLKG).

The exponential behavior and stabilizability of a stochastic 3D magnetohydrodynamic – Alpha model with cylindrical multiplicative noise

We study in this article, a stochastic evolution equation for the motion of magneto-fluid filling a periodic domain of : The existence and uniqueness of stationary solution is established and the long time behavior of solution is studied. The proof relies on a Galerkin approximation. Mainly, we prove that under some conditions on the forcing terms, the variational solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. We also prove a result related to the stabilization of the model.

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.

A free boundary problem of the cancer invasion

&lt;p style='text-indent:20px;'&gt;This paper deals with a free boundary problem for the cancer invasion model over a one dimensional habitat in the micro-environment, in which the free boundary represents the spreading front and is caused by tumour cells and acid-mediated. In this problem it is assumed that the tumour cells spread from the given initial region, and the spreading front expands at a speed that is proportional to the tumour cell and acids' population gradient at the front. The main objective is to realize the dynamics/variations of the healthy cells, tumour cells, acid-mediated and the free boundary. We prove a spreading-vanishing dichotomy for this model, namely the tumour cells either successfully spreads to infinity as time tends to infinite at the front, or it fails to establish and dies out in long run while the healthy cells stabilizes at a positive steady-state. The long time behavior of solution and criteria for spreading and vanishing are obtained.&lt;/p&gt;

Regularity results and asymptotic behavior for a noncoercive parabolic problem.

In this paper we study the regularity and the behavior in time of the solutions to a quasilinear class of noncoercive problems whose prototype is \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} u_{t} - \\mathrm{div}(a(x,t,u)\ abla u) = -\\mathrm{div}(u\\,E(x,t)) &{}\\quad \ ext{ in }\\, \\Omega \ imes (0,T), \\\\ u(x,t) = 0 &{}\\quad \ ext{ on }\\, \\partial \\Omega \ imes (0,T), \\\\ u(x,0) = u_{0}(x) &{}\\quad \ ext{ in }\\, \\Omega . \\end{array} \\right. \\end{aligned}$$\\end{document}ut-div(a(x,t,u)∇u)=-div(uE(x,t))inΩ×(0,T),u(x,t)=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ. In particular we show that under suitable conditions on the vector field E, even if the problem is noncoercive and although the initial datum u_0 is only an L^{1}(Omega ) function, there exist solutions that immediately improve their regularity and belong to every Lebesgue space. We also prove that solutions may become immediately bounded. Finally, we study the behavior in time of such regular solutions and we prove estimates that allow to describe their blow-up for t near zero.

Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

Long time behavior and turnpike solutions in mildly non-monotone mean field games

We consider mean field game systems in time-horizon (0,T), where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e.the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards (0,∞), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

On global dynamics of 2D convective Cahn\u2013Hilliard equation

In this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in H^{4}(Omega ) when the initial value belongs to H^{1}(Omega ).

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.

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.