Asymptotic behavior for coupled abstract evolution equations with one infinite memory

In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.     

Bresse system with infinite memories

In this paper, we consider a one‐dimensional linear Bresse system with infinite memories acting in the three equations of the system. We establish well‐posedness and asymptotic stability results for the system under some conditions imposed into the relaxation functions regardless to the speeds of wave propagations. Copyright © 2014 John Wiley & Sons, Ltd.     

Polynomial stability of solutions for a system of non-linear viscoelastic equations

In this article, we consider an interacting system consisting of two weakly-coupled viscoelastic equations. This work proves a uniform stabilization result without assuming any decay rate of the relaxation function provided that the kernels have some flat zones, the size of the flat zones are sufficiently small.     

Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay

In this article, we investigate the existence and asymptotic stability in p-th moment of a mild solution to a class of neutral stochastic integro-differential equation of fractional order involving non-instantaneous impulses with infinite delay in a Hilbert space. A new set of sufficient conditions proving existence and asymptotic stability of mild solution is derived by utilizing solution operator, functional analysis, stochastic analysis and fixed point technique. Finally, an example is provided to illustrate the obtained abstract result.     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form

     

Global asymptotic behavior of positive solutions for exponential form difference equations with three parameters

In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.     

Polynomial stability of a coupled system of wave equations weakly dissipative

In this work, we consider a coupled system of wave equation. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. We presented a class of examples of application of the main result.     

Qualitative theory for strongly damped wave equations

We investigate the asymptotic periodicity, L^p‐boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.     

First-order partial differential equations with large high-frequency terms

Systems of first-order semilinear partial differential equations with terms that oscillate at a frequency ω ? 1 in a single variable and are proportional to \(\sqrt \omega \) are considered. The Krylov-Bogolyubov-Mitropol’skii averaging method is substantiated for such equations. Based on the two-scale expansion method, an algorithm for constructing complete asymptotics of solutions is proposed and justified.     

On solutions of quasilinear wave equations with nonlinear damping terms

In this paper we consider the existence and asymptotic behavior of solutions of the following problem:

On infinite order and fully nonlinear partial differential evolution equations

In this paper we study the existence, uniqueness and propagation of regularity to infinite order partial differential evolution equations. Our approach is essentially functional and brings interesting results even when we restrict ourselves to finite order equations.     

Exp-function method for traveling wave solutions of nonlinear evolution equations

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations. The proposed technique is tested on the modified Zakharov-Kuznetsov (ZK) and Zakharov-Kuznetsov-Modified-Equal-Width (ZK-MEW) equations. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed exp-function method.     

General decay for a quasilinear system of viscoelastic equations with nonlinear damping

In this paper, we consider a system of coupled quasilinear viscoelastic equations with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.     

Asymptotic stability in thermoelectromagnetism with memory

A model describing a linear homogeneous dielectric with memory which obeys the Cattaneo–Maxwell law for the heat conduction is presented. The restrictions on the constitutive functionals are found as a direct consequence of the Second Law of Thermodynamics and some free energy potentials exhibited. Such potentials allow to determine a domain of dependence theorem for the first‐order integro‐differential system of equations governing the evolution of the thermoelectromagnetic radiation. The dissipativity due to the memory and to the heat conduction allows to establish some estimates on the asymptotic behaviour and prove the exponential decay of the solution of the system in absence of external sources. Copyright © 1999 John Wiley & Sons, Ltd.     

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.     

A general stability result for a nonlinear wave equation with infinite memory

In this work we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, among which those of exponential decay type, usually considered in the literature, are only special cases.     

Uniform and weak stability of Bresse system with two infinite memories

In this paper, we consider one-dimensional linear Bresse systems in a bounded open domain under Dirichlet–Neumann–Neumann boundary conditions with two infinite memories acting only on two equations. First, we establish the well-posedness in the sense of semigroup theory. Then, we prove two (uniform and weak) decay estimates depending on the speeds of wave propagations, the smoothness of initial data and the arbitrarily growth at infinity of the two relaxation functions.