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

Averaging of equations of viscoelasticity with singularly oscillating external forces

Given $\rho\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$\partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon )$$ together with the {\it averaged} equation $$\partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t).$$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,\tau)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors ${\mathcal A}^\varepsilon$. Within the further assumption $\rho<1$, the family ${\mathcal A}^\varepsilon$ turns out to be bounded in ${\mathcal H}$, uniformly with respect to $\varepsilon\in[0,1]$. The convergence of the attractors ${\mathcal A}^\varepsilon$ to the attractor ${\mathcal A}^0$ of the averaged equation as $\varepsilon\to 0$ is also established

Uniform attractors for non-autonomous wave equations with nonlinear damping

We consider dynamical behavior of non-autonomous wave-type evolutionary equations with nonlinear damping, critical nonlinearity, and time-dependent external forcing which is translation bounded but not translation compact (i.e., external forcing is not necessarily time-periodic, quasi-periodic or almost periodic). A sufficient and necessary condition for the existence of uniform attractors is established using the concept of uniform asymptotic compactness. The required compactness for the existence of uniform attractors is then fulfilled by some new a priori estimates for concrete wave type equations arising from applications. The structure of uniform attractors is obtained by constructing a skew product flow on the extended phase space for the norm-to-weak continuous process.Comment: 33 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

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)

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

A reaction-diffusion equation with memory

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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