Time Behavior Of Solutions Research Articles

Global attractor for a class of nonlinear generalized Kirchhoff models

The paper studies the long time behavior of solutions to the initial boundary value problem(IBVP) for a class of Kirchhoff models flow .We establish the well-posedness, theexistence of the global attractor in natural energy space

Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$

We consider a damped forced nonlinear Schrodinger-Poisson system in $R^3$. This provides us with an infinite-dimensional dynamical system in the energy space $H^1(R^3)$. We prove the existence of a finite dimensional global attractor for this dynamical system.

A nonlinear transport problem of hyperbolic-elliptic type

A nonlinear transport problem of hyperbolic-elliptic type is studied. Using estimates of potentials over varying domains and the method of characteristics, the initial value problem for Holder continuous data can be treated as an abstract evolution equation via Picard-Lindelof theorem. In addition, a global existence result is proved. Modifications of these techniques yield shock front solutionswith smooth interfaces and an extension to the case of only bounded initial values. Moreover, long time behavior of solutions and the relation to diffeomorphisms with prescribed Jacobian determinants are analyzed.

Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model

In this paper, we study the long time behavior of solution to the initial boundary value problem for a class of Kirchhoff-Boussinesq model flow . We first prove the wellness of the solutions. Then we establish the existence of global attractor.

Behaviour in time of solutions to a class of fourth order evolution equations

We consider some initial-boundary value problems for a class of nonlinear parabolic equations of the fourth order, whose solution u(x,t) may or may not blow up in finite time. Under suitable conditions on data, a lower bound for t⁎ is derived, where [0,t⁎) is the time interval of existence of u(x,t). Under appropriate assumptions on the data, a criterion which ensures that u cannot exist for all time is given, and an upper bound for t⁎ is derived. Some extensions for a class of nonlinear fourth order parabolic systems are indicated.

Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients

In this article we investigate two free boundary problems for a Lotka-Volterra competition system in a higher space dimension with sign-changing coefficients. One may be viewed as describing how two competing species invade if they occupy an initial region, the other describes the dynamical process of a new competitor invading into the habitat of a native species. For simplicity, it is assumed that the environment is radially symmetric. The main purpose of this article is to understand the asymptotic behavior of competing species spreading via a free boundary. We derive some sufficient conditions for species spreading success and spreading failure. Moreover, when spreading successfully, we provide the long time behavior of solutions.

A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries

In this paper we investigate a free boundary problem for a predator-prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading-vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as $t\to \infty$ at both fronts and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.

Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience

The Noisy Integrate-and-Fire equation is a standard non-linear Fokker–Planck equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b < 0 inhibitory networks. In the excitatory case, it was proved that, once the proportion of neurons that are close to their action potential is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincaré inequalities leading to L2 estimates and on the other hand, on the notion of ‘universal super-solution’ and parabolic regularizing effects to obtain bounds.

Global existence and long time behavior of solutions of a quantum Boltzmann equation

In this paper, we study a quantum Boltzmann equation with a harmonic oscillator for isotropic gases of bosons and fermions, respectively. This model comes from physics literatures (see, e.g., [M. Holland, J. Williams and J. Cooper, Bose–Einstein condensation: Kinetic evolution obtained from simulated trajectories, Phys. Rev. A 55 (1997) 3670–3677]). The distribution function, i.e. the solution, is discrete in the energy variable. We give the classification of equilibria of the equation for bosons and fermions, respectively, and prove the global existence, uniqueness and the strong convergence to equilibrium for solutions of the equation.

Metastability for parabolic equations with drift: part 1

We provide a self-contained analysis, based entirely on pde methods, of the exponentially long time behavior of solutions to linear uniformly parabolic equations which are small perturbations of a transport equation with vector field having a globally stable point. We show that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. This work extends previous results of Freidlin and Wentzell and Freidlin and Koralov and applies also to semilinear elliptic pde.

Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term

We investigate the boundedness and large time behavior of solutions of the Cauchy-Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: $$ u_t = (|u-x|^{p-2} u-x)_x+|u_x|^q \qquad \text{in}\quad (0, +\infty)\tiles(0, 1),\qquad q > p > 2.$$ We prove that: either $u_x$ blows up in finite time, or $u$ is global and converges in $W^{1, \infty}$norm to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.

Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay

We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay. We investigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force depending on the velocity of fluid on a bounded domain. By a prior estimate and a contractive function, we give a sufficient condition for the existence of pullback attractor of NSV equation.

Long Time Decay Rate to a Bipolar Quantum Drift-Diffusion Model

This paper is devoted to studying the long time behavior of solutions to a bipolar quantum hydrodynamic in one-dimensional space for general pressure functions. The model is usually applied to simulate some quantum effects in semiconductor devices. The decay rate for time variable is obtained by the entropy functional method and semidiscrete technique.

Final State Problem for the Cubic Nonlinear Schrödinger Equation with Repulsive Delta Potential

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given small asymptotic profile u ap , there exists a solution u to (δ-NLS) which converges to u ap in L 2(ℝ) as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.

Large time behavior of solutions for the porous medium equation with a nonlinear gradient source

Abstract This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source u t = Δ u m + | ∇ u l | q , ( x , t ) ∈ Ω × ( 0 , ∞ ) , where l ≥ m &gt; 1 and 1 ≤ q &lt; 2 . When l q = m , we prove that the global solution converges to the separate variable solution t − 1 m − 1 f ( x ) . While m &lt; l q ≤ m + 1 , we note that the global solution converges to the separate variable solution t − 1 m − 1 f 0 ( x ) . Moreover, when l q &gt; m + 1 , we show that the global solution also converges to the separate variable solution t − 1 m − 1 f 0 ( x ) for the small initial data u 0 ( x ) , and we find that the solution u ( x , t ) blows up in finite time for the large initial data u 0 ( x ) . MSC:35K55, 35K65, 35B40.

Green’s function of a chemotaxis model in the half space and its applications

In this paper, we are interested in a degenerate Keller–Segel model in the half space xn>lt in high dimensions, with a constant l. Under the conservative boundary condition, we study the global solvability, regularity and long time behavior of solutions. We first construct Green’s function of the half space problem and study its properties. Then by Green’s function, we give the implicit formula of solutions. We prove that when the initial data is small enough, the half space problem is always globally and classically solvable. We also obtain decay estimates of solutions when the parameter l have different signs. For l<0, the time decay of solutions is (1+t)−n/2. While for l>0, the time decay is (1+t)−(n−1)/2, and in this case, we also obtain an exponential decay in the spatial space in the xn-direction.

Long time behavior of solutions to coupled Burgers-complex Ginzbury-Landau (Burgers-CGL) equations for flames governed by sequential reaction

This paper studies the existence and long time behavior of the solutions to the coupled Burgers-complex Ginzburg-Landau (Burgers-CGL) equations, which are derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction, having two flame fronts corresponding to two reaction zones with a finite separation distance between them. This paper firstly shows the existence of the global solutions to these coupled equations via subtle transforms, delicate a priori estimates and a so-called continuity method, then prove the existence of the global attractor and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor.

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.

Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics

In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions. Such scaling argument is very common for the study of long time behavior of solutions to evolutionary problems where a scaling invariance of the main part of the operator is present.

H 2-regularity random attractors of stochastic non-Newtonian fluids with multiplicative noise

In this paper, the authors study the long time behavior of solutions to stochastic non-Newtonian fluids in a two-dimensional bounded domain, and prove the existence of H2-regularity random attractor.