A New Approach to Equations with Memory

In this work, we present a novel approach to the mathematical analysis of equations with memory based on the notion of a state, namely, the initial configuration of the system which can be unambiguously determined by the knowledge of the future dynamics. As a model, we discuss the abstract version of an equation arising from linear viscoelasticity. It is worth mentioning that our approach goes back to the heuristic derivation of the state framework, devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state in viscoelasticity: new free energies and applications to PDEs", Arch. Ration. Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical motivations, we develop a suitable functional formulation which, as far as we know, is completely new.Comment: 39 pages, no figur

Stability of abstract linear thermoelastic systems with memory

An abstract linear thermoelastic system with memory is here considered. Existence, uniqueness, and continuous dependence results are given. In presence of regular and convex memory kernels, the system is shown to be exponentially stable. An application to the Kirchhoff plate equation is given

Asymptotic behavior of a nonlinear hyperbolic heat equation with memory

n this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an integro-differential equation describing the heat flow in a rigid heat conductor with memory. This model arises matching the energy balance, in presence of a nonlinear time-dependent heat source, with a linearized heat flux law of the Gurtin-Pipkin type. Existence and uniqueness of solutions for the corresponding semilinear system (subject to initial history and Dirichlet boundary conditions) is provided. Moreover, under proper assumptions on the heat flux memory kernel and the magnitude of nonlinearity, the existence of a uniform absorbing set is achieved

Uniform attractors for a phase-field model with memory and quadratic nonlinearity

A phase-field system with memory which describes the evolution of both the temperature variation $\theta$ and the phase variable $\chi$ is considered. This thermodynamically consistent model is based on a linear heat conduction law of Coleman-Gurtin type. Moreover, the internal energy linearly depends both on the present value of $\theta$ and on its past history, while the dependence on $\chi$ is represented through a function with quadratic nonlinearity. A Cauchy-Neumann initial and boundary value problem associated with the evolution system is then formulated in a history space setting. This problem is shown to generate a non-autonomous dynamical system which possesses a uniform attractor. In the autonomous case, the attractor has finite Hausdorff and fractal dimensions whenever the internal energy linearly depends on $\chi$

Uniform attractors for a non-autonomous semilinear heat equation with memory

n this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of a non-autonomous integro-partial differential equation describing the heat how in a rigid heat conductor with memory. Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of uniform absorbing sets and of a global uniform attractor is achieved. In the case of quasiperiodic dependence of time of the external heat supply the above attractor is shown to have finite Hausdorff dimension

Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

In this paper the long time behaviour of the solutions of 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega) and then it is proved that this global attractor is a bounded subset of H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)

Exponential stability of the wave equation with memory and time delay

We study the asymptotic behaviour of the wave equation with viscoelastic damping in presence of a time-delayed damping. We prove exponential stability if the amplitude of the time delay term is small enough

Global Attractors for an Extensible Thermoelastic Beam System

This work is focused on the dissipative system describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of the temperature. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When both the external body force and the heat source are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all values of the external axial load. The same result holds true when the rotational inertia is taken into consideration. In both cases, the solutions on the attractor are strong solutions.Comment: 21 pages, no figur