Long-time Behavior Of Solutions Research Articles

Remark on global attractor for damped wave equation on R 3 $\mathbb{R}^{3}$

This paper is concerned with the long-time behavior of solutions to the weakly damped wave equation with lower regular forcing defined on the entire space $\mathbb{R}^{3}$ . The existence of a global attractor in $H^{1}(\mathbb{R}^{3}) \times L^{2}(\mathbb{R}^{3})$ is proven. Moreover, under some additional condition, the translational regularity of the attractor is established.

Long time behavior of solutions to the mKdV

ABSTRACTIn this paper we consider the long time behavior of solutions to the modified Korteweg-de Vries equation on ℝ. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotics without relying on complete integrability. We also consider the asymptotic completeness problem. Our result uses the method of testing by wave packets, developed in the work of Ifrim and Tataru on the 1d cubic nonlinear Schrödinger and 2d water wave equations.

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.

Asymptotic behavior of solutions of a reaction–diffusion equation with inhomogeneous Robin boundary condition and free boundary condition

This paper studies the long time behavior of solutions of a reaction–diffusion model with inhomogeneous Robin boundary condition at x=0 and free boundary condition at x=h(t). We prove that, for the initial data u0=σϕ, there exists σ∗⩾0 such that u(⋅,t) converges to a positive stationary solution which tends to 1 as x→∞ locally uniformly in [0,∞) when σ>σ∗. In the case of σ⩽σ∗ the solution u(⋅,t) converges to the ground state V(⋅−z) where V is the unique even positive solution of V″+f(V)=0 subject to V(∞)=0 and z is the root of aV′(−z)−(1−a)V(−z)=b. The asymptotic behavior of the solutions is quite different from the homogeneous case b=0.

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.

Long-Time Behavior of Solutions to a Class of Parabolic Equations with Nonstandard Growth Condition

In this paper, the author studies the long-time behavior of solutions to the p(x)-Laplace equation \(u_t=\mathrm {div}(|\nabla u|^{p(x)-2}\nabla u)+f(u)\), with homogeneous Dirichlet boundary condition in a bounded domain. As for the blow-up results, it is shown, by using the energy method, that the solutions of this problem blow up in finite time for nonpositive initial energy, or even for small positive initial energy. As for the extinction results, we give some sufficient conditions for the solutions to vanish in finite time. All these results generalize the ones when p(x) is a constant.

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.

Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries

We consider Fisher-KPP equation with advection: u t = u x x − β u x + f ( u ) for x ∈ ( g ( t ) , h ( t ) ) , where g ( t ) and h ( t ) are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient − β on the long time behavior of the solutions. We find two parameters c 0 and β ⁎ with β ⁎ > c 0 > 0 which play key roles in the dynamics, here c 0 is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data { σ ϕ } σ > 0 (where ϕ is some compactly supported positive function), we show that: (1) in case β ∈ ( 0 , c 0 ) , there exists σ ⁎ ⩾ 0 such that spreading happens when σ > σ ⁎ (i.e., u ( t , ⋅ ; σ ϕ ) → 1 locally uniformly in R ) and vanishing happens when σ ∈ ( 0 , σ ⁎ ] (i.e., [ g ( t ) , h ( t ) ] remains bounded and u ( t , ⋅ ; σ ϕ ) → 0 uniformly in [ g ( t ) , h ( t ) ] ); (2) in case β ∈ ( c 0 , β ⁎ ) , there exists σ ⁎ > 0 such that virtual spreading happens when σ > σ ⁎ (i.e., u ( t , ⋅ ; σ ϕ ) → 0 locally uniformly in [ g ( t ) , ∞ ) and u ( t , ⋅ + c t ; σ ϕ ) → 1 locally uniformly in R for some c > β − c 0 ), vanishing happens when σ ∈ ( 0 , σ ⁎ ) , and in the transition case σ = σ ⁎ , u ( t , ⋅ + o ( t ) ; σ ϕ ) → V ⁎ ( ⋅ − ( β − c 0 ) t ) uniformly, the latter is a traveling wave with a “big head” near the free boundary x = ( β − c 0 ) t and with an infinite long “tail” on the left; (3) in case β = c 0 , there exists σ ⁎ > 0 such that virtual spreading happens when σ > σ ⁎ and u ( t , ⋅ ; σ ϕ ) → 0 uniformly in [ g ( t ) , h ( t ) ] when σ ∈ ( 0 , σ ⁎ ] ; (4) in case β ⩾ β ⁎ , vanishing happens for any solution.

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.

Long time behavior of solutions to the compressible MHD system in multi-dimensions

In this paper, we consider the initial value problem for the compressible MHD system in multi-dimensions. Under suitable conditions on the initial value, the global existence and long time behavior of classical solutions are established. Moreover, we also show that the solution can be approximated by the linear solution as time tends to infinity. The proof is based on the decay properties of solution operator to the compressible MHD system, which may be given in terms of the heat kernel and the solution operator to a linear wave equation that is corresponding to the density function. The decay properties of solution operators to the compressible MHD system may be derived from the pointwise estimate of solution operator to the linear wave equation. Finally, asymptotic profile to the linear system is also established.

The minimal habitat size for spreading in a weak competition system with two free boundaries

In this paper, we focus on the dynamics for a Lotka–Volterra type weak competition system with two free boundaries, where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species, respectively. In the weak competition case, the dynamics of this model can be classified into four cases, which forms a spreading–vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Some sufficient conditions for spreading and vanishing are established. Also, when spreading occurs, some rough estimates for spreading speed and the long-time behavior of solutions are established.

On an ODE-PDE coupling model of the mitochondrial swelling process

Mitochondrial swelling has huge impact to multicellular organisms since it triggers apoptosis,the programmed cell death. In this paper we present a new mathematical model of this phenomenon. As a novelty it includes spatial effects, which are of great importance for the in vivo process. Our model considers three mitochondrial subpopulations varying in the degree of swelling. The evolution of these groups is dependent on the present calcium concentration and is described by a system of ODEs, whereas the calcium propagation is modeled by a reaction-diffusion equation taking into account spatial effects. We analyze the derived model with respect to existence and long-time behavior of solutions and obtain a complete mathematical classification of the swelling process.

On the Transfer of Energy Towards Infinity in the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation

We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schr\"odinger equation. In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e. a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a self-similar manner.

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.

Global attractor for weakly damped gKdV equations in higher sobolev spaces

Long time behavior of solutions for weakly damped gKdV equations on the real line is studied. With some weak regularity assumptions on the force $f$, we prove the existence of global attractor in $H^s$ for any $s\geq 1$. The asymptotic compactness of solution semigroup is shown by Ball's energy method and Goubet's high-low frequency decomposition if $s$ is an integer and not an integer, respectively.

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.

The reaction-diffusion system for an SIR epidemic model with a free boundary

The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.

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.

A General Stability Result for Viscoelastic Equations with Singular Kernels

Viscoelasticity with regular relaxation functions has attracted the attention of many researchers over the last half century or so. Several results concerning existence and long-time behavior of solutions have been established. In particular the exponential, polynomial decay and recently what so called the general decay have been proved. For viscoelasticity, with singular kernels, less attention has been given and few results of existence and exponential decay have been established. In this paper we extend the general decay result, established for regularkernel viscoelasticity, to that with singular kernels. We also present some numerical test to illustrate our theoretical result.