Long-time Behavior Of Solutions Research Articles

Uniform attractors for impulsive reaction–diffusion equations

The goal of this paper is to consider the long time behavior of solutions of reaction–diffusion equations with impulsive effects at fixed moment of time. Under a new class of impulse function, we prove the existence of uniform attractors in the spaces L 2 ( Ω ) , H 0 1 ( Ω ) , L p ( Ω ) and L 2 p - 2 ( Ω ) , respectively.

Ionization of Coulomb Systems in $${\mathbb{R}^3}$$ by Time Periodic Forcings of Arbitrary Size

We analyze the long time behavior of solutions of the Schrodinger equation $${i\psi_t=(-\Delta-b/r+V(t,x))\psi}$$ , $${x\in\mathbb{R}^3}$$ , r = |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) = V(t + 2π/ω, x) with zero time average. We show that, for any V(t, x) of the form $${2\Omega(r) \sin (\omega t-\theta)}$$ , with Ω(r) nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, i.e. $${P(t, K) = \int_K|\psi(t,x)|^2dx\to 0}$$ as t → ∞ for any compact set $${K\subset\mathbb{R}^3}$$ . Furthermore, if the initial state is compactly supported and has only finitely many spherical harmonic modes, then P(t, K) decays like $${t^{-5/3}}$$ as t → ∞. To prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.

Long-time behaviour of solutions of a non-linear diffusion problem with non-local source term

Long-time behaviour of solutions of a non-linear diffusion problem with non-local source term

Long-Time Behavior of Solutions to a Class of Forward-Backward Parabolic Equations

We consider weak entropy measure-valued solutions of the Neumann initial-boundary value problem for the equation $u_t=[\phi(u)]_{xx}$, where $\phi$ is nonmonotone. These solutions are obtained from the corresponding problem for the regularized equation $u_t=[\phi(u)]_{xx}+\varepsilon u_{xxt}$ ($\varepsilon>0$) by a vanishing viscosity method and satisfy a family of suitable entropy inequalities. Relying on a strong version of these inequalities, we prove exhaustive results concerning the long-time behavior of solutions.

Dynamics of a structured neuron population

We study the dynamics of assemblies of interacting neurons. For large fully connected networks, the dynamics of the system can be described by a partial differential equation reminiscent of age-structure models used in mathematical ecology, where the ‘age’ of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network.We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand, we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behaviour of solutions; both for small and large J, we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing.

Exact equilibrium solutions of a diffusion equation with a nonlinear diffusion term by means of Jacobian elliptic functions

We construct exact equilibrium solutions of a diffusion equation with a nonlinear diffusion term by means of Jacobian elliptic functions, the direct method, and an f 4 model through a simple mapping relation. In the limit, as the elliptic modulus of the Jacobian elliptic function tends to 0 and to 1, we deduce the trigonometric and solitary wave solutions, respectively. The long-time behaviour of solutions is investigated via numerical simulations.

The influence of damping and source terms on solutions of nonlinear wave equations

We discuss in this paper some recent development in the study of nonlinear wave equations. In particular, we focus on those results that deal with wave equations that feature two competing forces.One force is a damping term and the other is a strong source. Our central interest here is to analyze the influence of these forces on the long-time behavior of solutions.

Remarks on bounded solutions of steady Hamilton–Jacobi equations

We study here the equation H ( D u ) = H ( 0 ) , x ∈ R N . More precisely we investigate under which hypotheses the constant functions are the only bounded solutions. In arbitrary space dimension we prove that this happens when convexity and coercivity occur. In one space dimension we show that the above property holds true for Hamiltonians in a larger class. These results apply when studying the long time behaviour of solutions for time-dependent Hamilton–Jacobi equations. To cite this article: M. Bostan, G. Namah, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Pullback attractors for non-autonomous reaction-diffusion equations on ℝ n

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝn when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L2(ℝn) and H1(ℝn), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.

A Bottle in a Freezer

We propose here a model for solidification of the liquid content of an elastic bottle in a freezer. The main goal is to explain the occurrence of high stresses inside the bottle. As a by-product, we derive a formula for the undercooling coefficient in terms of the elasticity constants, latent heat, and the phase expansion coefficient. We investigate the well-posedness of the three-dimensional model: we prove the existence and uniqueness of a solution for the corresponding initial-boundary value problem which couples a PDE with an integrodifferential equation and an ordinary differential inclusion ruling the evolution of the phase parameter. Finally, we prove some results on the long time behavior of solutions.

Existence of Global Attractors in for -Laplacian Parabolic Equation in

We study the long-time behavior of solution for the -Laplacian equation in , in which the nonlinear term is a function like with , , or with and . We prove the existence of a global -attractor for any .

Some notes on a nonlinear degenerate parabolic equation

In this note we study nonexistence and long time behavior of solutions for a nonlinear degenerate parabolic equation of non-divergence type. In addition, we also construct some special explicit solutions which may blow up in finite time.

A Note on nonautonomous klein-gordon-schrödinger equations with homogeneous dirichlet boundary condition

This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrödinger equations with homogeneous Dirichlet boundary condition. The authors prove the existence of compact kernel sections for the associated process by using a suitable decomposition of the equations.

Asymptotic behavior of solutions to a boundary value problem for a nonlinear equation with a fractional derivative

This paper analyzes the long-time behavior of solutions to a boundary value problem on a half-line for the following nonlinear nonlocal model evolution equation with a fractional derivative:

On the Dynamics of Controlled Magnetohydrodynamic Systems

In this paper we study the long time behavior of solutions for an optimal control problem associated with the viscous incompressible electrically conducting fluid modeled by the magnetohydrodynamic (MHD) equations in a bounded two dimensional domain through the adjustment of distributed controls. We first construct a quasi-optimal solution for the MHD systems which possesses exponential decay in time. We then derive some preliminary estimates for the long-time behavior of all admissible solutions of the MHD systems. Next we prove the existence of a solution for the optimal control problem for both finite and infinite time intervals. Finally, we establish the long-time decay properties of the solutions for the optimal control problem.

DIFFUSION AND BIFURCATION PROBLEMS IN SINGULARLY PERTURBED DOMAINS

. Diffusion problems under singular perturbations of the domain or the boundary conditions are analyzed. The first problem that we consider is the diffusion of a material from a domain that is nearly impermeable, having only several small patches on the boundary where the material can slowly leak out. The second problem that is studied is the diffusion of a material that originates from some localized regions in a two or three-dimensional domain. Steady-state solutions and the long-time behavior of solutions are analyzed in detail. Finally, the analysis is extended to determine the change in bifurcation values associated with nonlinear diffusion equations under singular perturbations of the domain. The results are then applied to a model in resource management.

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hperk is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hperk.

Uniform attractor for a two-dimensional nonautonomous incompressible non-Newtonian fluid

We discuss the long time behavior of solutions for a two-dimensional (2D) nonautonomous incompressible non-Newtonian fluid in 2D bounded domains. When the external force g 0 ( x , t ) is translation compact (tr.c.) in L loc 2 ( R ; H ) , we obtain the existence and reveal the structure of the uniform attractor for the processes associated to the fluid by constructing skew product flow on the extended phase space. When ‖ g 0 ‖ L b 2 is properly small, we establish that there exists a unique bounded asymptotically stable solution to the fluid.

The global attractors for the dissipative generalized Hasegawa-Mima equation

The long time behavior of solutions of the generalized Hasegawa-Mima equation with dissipation term is considered. The existence of global attractors of the periodic initial value problem is proved, and the estimate of the upper bound of the Hausdorff and fractal dimensions for the global attractors is obtained by means of uniform a priori estimates method.

Compact uniform attractors for dissipative lattice dynamical systems with delays

In this paper, weconsider the long time behavior of solutions for dissipative latticedynamical systems with delays. We first prove a sufficient andnecessary condition for the existence of a compact uniform attractorfor the family of processes corresponding to the lattice dynamicalsystems with delays. Then we apply this result to prove theexistence of a compact uniform attractor for the process associatedto the retarded lattice Zakharov equations. As a consequence, someresults for the non-delay lattice dynamical systems are deduced asparticular cases.