Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain

This paper is devoted to the study of the asymptotic behavior of the solutions of singularly perturbed transmission problems in a periodically perforated domain. The domain is obtained by making in a periodic set of holes, each of them of size proportional to a positive parameter ε. We first consider an ideal transmission problem and investigate the behavior of the solution as ε tends to 0. In particular, we deduce a representation formula in terms of real analytic maps of ε and of some additional parameters. Then we apply such result to a nonideal nonlinear transmission problem.     

Asymptotic behavior of solutions of a periodic diffusion system

This paper deals with a periodic reaction-diffusion system of plankton allelopathy under homogeneous Neumann boundary conditions. Based on the result of Ahmad and Lazer, we show some estimates and nonexistence results for the positive solutions of the system. Furthermore, we investigate the asymptotic behavior of the solutions of the system, that is one species dies out and the other exists as time t tends to infinity.     

Asymptotic solution of nonlinear nonlocal singularly perturbed reaction diffusion problems with two parameters

A class of differential-difference reaction diffusion equations initial boundary problem with a small time delay is considered. Under suitable conditions and by using method of the stretched variable, the formal asymptotic solution is constructed. And then, by using the theory of differential inequalities the uniformly validity of solution is proved.     

Asymptotic behaviour of some nonlocal diffusion problems

We deal with a parabolic equation having a diffusion coefficient depending on a nonlocal quantity. We investigate the convergence of the solution towards a steady state, extending previous results obtained by [M. Chipot and B. Lovat (1999). On the asympotic behaviour of some nonlocal problems. Positivity, 3, 65-81]. Using the dynamical systems point of view, we are able to treat the case of a continuum of steady states     

Asymptotic behavior of maximum likelihood estimators in a branching diffusion model

In this paper the consistency and asymptotic normality of maximum-likelihood estimations for a super-critical branching diffusion model are obtained under certain conditions on its drift, variance and reproduction law. We proceeded by first studying the limit behavior of the Fisher information measure and related processes, and then verifying conditions established in Barndorff-Nielsen and Sørensen (Int stat Rev 62:133–165, 1994). This in turn uses the Martingale Law of Large Numbers as well as the Martingale Central Limit Theorem.     

Asymptotic dynamics for reaction diffusion equations in unbounded domain

In this paper we study the asymptotic dynamics for reaction diffusion equation defined in $\mathbb{R}^n$. We will prove that the equation possesses a fixed point when the nonlinearity satisfies some restrictive conditions and then we show that the fixed point is an exponential attractor.     

Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm-Liouville problems defined on [0,π] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x=π, the asymptotic expressions associated with other boundary conditions are provided.     

Asymptotic behavior of solutions to the perturbed simple pendulum problems with two parameters

We consider the perturbed simple pendulum equation –u ″(t) + μ |u (t)|^{p –1}u (t) = λ sin u (t), t ∈ I ? (–T, T), u (t) > 0, t ∈ I, u (±T) = 0, where p > 1 is a constant,λ > 0 and μ ∈ R are parameters. The purpose of this paper is to clarify the structure of the solution set. To do this, we study precisely the asymptotic shape of the solutions when λ ? 1 as well as the asymptotic behavior of variational eigenvalue μ (λ) as λ → ∞. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation

We study the asymptotic behavior as t → ∞ of the solution of the initial-boundary value problem for the nonlinear integro-differential equation

$$\frac{{\partial U}}{{\partial t}} = \frac{\partial }{{\partial x}}\left[ {a\left( {\mathop \smallint \limits_0^t \left( {\frac{{\partial U}}{{\partial x}}} \right)^2 d\tau } \right)\frac{{\partial U}}{{\partial x}}} \right],$$

where a(S) = (1 + S) ^p , 0 < p ≤ 1. We consider problems with homogeneous boundary conditions as well as with a nonhomogeneous boundary condition on part of the boundary. The orders of convergence are established.

     

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.     

Asymptotic behaviour of equicoercive diffusion energies in dimension two

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F _n , , defined in L ²(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F _n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F _n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.     

Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy

This paper is concerned with the existence and asymptotic behavior of periodic solutions for a periodic reaction diffusion system of a planktonic competition model under Dirichlet boundary conditions. The approach to the problem is by the method of upper and lower solutions and the bootstrap argument of Ahmad and Lazer. It is shown under certain conditions that this system has positive or semi-positive periodic solutions. A sufficient condition is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Asymptotic behavior of SU(3) Toda system in a bounded domain

We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function.     

Asymptotic behavior for a class of the renewal nonlinear equation with diffusion

In this paper, we consider nonlinear age‐structured equation with diffusion under nonlocal boundary condition and non‐negative initial data. More precisely, we prove that under some assumptions on the nonlinear term in a model of McKendrick–Von Foerster with diffusion in age, solutions exist and converge (long‐time convergence) towards a stationary solution. In the first part, we use classical analysis tools to prove the existence, uniqueness, and the positivity of the solution. In the second part, using comparison principle, we prove the convergence of this solution towards the stationary solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains

In this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain. The functional setting is based on standard Lebesgue and Sobolev–Lebesgue spaces. We show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor. In particular cases, our results allow us to improve some previous known results.