Long-time Behavior Of Solutions Research Articles

Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion

This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space R3. We establish that, in the inviscid resistive case, the energy ‖b(t)‖22 vanishes and ‖u(t)‖22 converges to a constant as time tends to infinity provided the velocity is bounded in W1−α,3α(R3); in the viscous non-resistive case, the energy ‖u(t)‖22 vanishes and ‖b(t)‖22 converges to a constant provided the magnetic field is bounded in W1−β,∞(R3). In summary, one single diffusion, being as weak as (−Δ)αb or (−Δ)βu with small enough α,β, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.

The L2 decay for the 2D co-rotation FENE dumbbell model of polymeric flows

In this paper we mainly study the long time behavior of solutions to the finite extensible nonlinear elastic (FENE) dumbbell model with dimension two in the co-rotation case. Firstly, we obtain the L2 decay rate of the velocity of the 2D co-rotation FENE model is (1+t)−12 with small data. Then, by virtue of the Littlewood–Paley theory, we can remove the small condition. Our obtained sharp result improves considerably the recent results in [8,14].

Long-time behavior of solutions for the compressible quantum magnetohydrodynamic model in $${\mathbb {R}}^3$$ R 3

In this paper, long-time behavior of solutions for the compressible viscous quantum magnetohydrodynamic model in three-dimensional whole space is studied. We establish the optimal time decay rates for higher-order spatial derivatives of density, velocity and magnetic field, which improve the work of Pu and Xu (Z Angew Math Phys 68:18, 2017).

On the existence and long-time behavior of solutions to stochastic three-dimensional Navier–Stokes–Voigt equations

ABSTRACTWe consider the 3D stochastic Navier–Stokes–Voigt equations in bounded domains with the homogeneous Dirichlet boundary condition and infinite-dimensional Wiener process. First, we prove the existence and uniqueness of solutions to the problem. Then we investigate the mean square exponential stability and the almost sure exponential stability of the stationary solutions.

A two-species weak competition system of reaction-diffusion-advection with double free boundaries

<p style='text-indent:20px;'>In this paper, we investigate a two-species weak competition system of reaction-diffusion-advection with double free boundaries that represent the expanding front in a one-dimensional habitat, where a combination of random movement and advection is adopted by two competing species. The main goal is to understand the effect of small advection environment and dynamics of the two species through double free boundaries. We provide a spreading-vanishing dichotomy, which means that both of the two species either spread to the entire space successfully and survive in the new environment as time goes to infinity, or vanish and become extinct in the long run. Furthermore, if the spreading or vanishing of the two species occurs, some sufficient conditions via the initial data are established. When spreading of the two species happens, the long time behavior of solutions and estimates of spreading speed of both free boundaries are obtained.

A class of fourth-order parabolic equation with arbitrary initial energy

In this paper we apply the modified potential well method to the study of the long time behaviors of solutions to a class of fourth-order parabolic equation in a bounded smooth domain of Rn for arbitrary n≥1. Global existence and blow up in finite time of solutions are obtained when the initial data satisfy different conditions. To be a little more precise, we give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. Moreover, the decay rate of the L2 norm is also obtained for global solutions. Sufficient conditions for the existence of global and blow-up solutions are also provided for supercritical initial energy. These improve and generalize some recent results.

Energy Scattering Theory for Electromagnetic NLS in Dimension Two

We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schr¨odinger equations with scaling critical magnetic potentials in dimension two. In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov–Bohm potential which decays likely the Coulomb potential |x|−1.

Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy

In this paper we will apply the modified potential well method and variational method to the study of the long time behaviors of solutions to a class of parabolic equation of Kirchhoff type. Global existence and blow up in finite time of solutions will be obtained for arbitrary initial energy. To be a little more precise, we will give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. The decay rate of the L2(Ω) norm is also obtained for global solutions in these cases. Moreover, some sufficient conditions for the existence of global and blow-up solutions are also derived when the initial energy is supercritical.

Analysis of a System Describing Proliferative-Quiescent Cell Dynamics

Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis. Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations.

On the dynamics of an intraguild predator–prey model

An intraguild predator–prey model with a carrying capacity proportional to the biotic resource, is generalized by introducing a Holling type II functional response. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. The existence of biologically meaningful equilibria (boundary and internal equilibria) has been investigated. Linear and nonlinear stability conditions for biologically meaningful equilibria are performed. Finally, numerical simulations on different regimes of coexistence and extinction of the involved populations have been shown.

Existence of invariant measures for the stochastic damped KdV equation

We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.

Cloaking via Mapping for the Heat Equation

This paper explores the concept of near-cloaking in the context of time-dependent heat propagation. We show that after the lapse of a certain threshold time, the boundary measurements for the homogeneous heat equation are close to the cloaked heat problem in a certain Sobolev space norm irrespective of the density-conductivity pair in the cloaked region. A regularized transformation media theory is employed to arrive at our results. Our proof relies on the study of the long time behavior of solutions to the parabolic problems with high contrast in density and conductivity coefficients. It further relies on the study of boundary measurement estimates in the presence of small defects in the context of steady conduction problems. We then present some numerical examples to illustrate our theoretical results.

On the time evolution of Bernstein processes associated with a class of parabolic equations

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved.

Long-time behavior of the one-phase Stefan problem in periodic and random media

We study the long-time behavior of solutions of the one-phase Stefan problem in inhomogeneous media in dimensions n ≥ 2. Using the technique of rescaling which is consistent with the evolution of the free boundary, we are able to show the homogenization of the free boundary velocity as well as the locally uniform convergence of the rescaled solution to a self-similar solution of the homogeneous Hele-Shaw problem with a point source. Moreover, by viscosity solution methods, we also deduce that the rescaled free boundary uniformly approaches a sphere with respect to Hausdorff distance.

Complex Ginzburg–Landau equations with dynamic boundary conditions

The initial-dynamic boundary value problem (idbvp) for the complex Ginzburg–Landau equation (CGLE) on bounded domains of RN is studied by converting the given mathematical model into a Wentzell initial–boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behavior of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.

Long-time behavior of a class of nonlocal partial differential equations

This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain \begin{document}$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$ \end{document} Firstly, due to the lack of an upper growth restriction of the nonlinearity $f$, we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of \begin{document}$(L^2_0(Ω), L^2_0(Ω))$\end{document} -absorbing sets and \begin{document}$(L^2_0(Ω), H^{σ/2}_0(Ω))$\end{document} -absorbing sets for the solution semigroup \begin{document}$\{S(t)\}_{t≥q 0}$\end{document} . Finally, we prove the existence of \begin{document}$(L^2_0(Ω), L^2_0(Ω))$\end{document} -global attractor and \begin{document}$(L^2_0(Ω), H^{σ/2}_0(Ω))$\end{document} -global attractor by a asymptotic compactness method.

On the nonlinear dynamics of an ecoepidemic reaction–diffusion model

A reaction–diffusion ecoepidemic model of predator–prey type with a transmissible disease spreading among the predator species only is considered. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. Conditions guaranteeing the non existence of non-constant equilibria have been found. Linear and non-linear stability conditions for biologically meaningful equilibria are determined.

The Global and Exponential Attractors for the Higher-order Kirchhoff-type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: ${u_{tt}} + {( - \Delta )^m}{u_t} + {\left( {\alpha + \beta\left\| {{\nabla ^m}u} \right\|^2} \right)^{q}}{( - \Delta )^m}u + g(u) = f(x)$. At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in $H_0^m(\Omega ) \times {L^2}(\Omega )$ according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.

Existence of global attractor for the one-dimensional bipolar quantum drift-diffusion model

In this paper, we investigate a one-dimensional bipolar quantum drift-diffusion model from semiconductor devices. We mainly show the long-time behavior of solutions to the one-dimensional bipolar quantum drift-diffusion model in a bounded domain. That is, we prove the existence of the global attractor for the solution.

Long-time behavior of solutions and chaos in reaction-diffusion equations

It is shown that members of a class (of current interest with many applications) of non-dissipative reaction-diffusion partial differential equations with local nonlinearity can have an infinite number of different unstable solutions traveling along an axis of the space variable with varying speeds, traveling impulses and also an infinite number of different states of spatio-temporal (diffusion) chaos. These solutions are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and, as an example, it is explained how space-time chaos can arise. Results of the same type are also obtained in the case of a nonlocal nonlinearity.