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We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation $$ u_t - u_{txx} + u_x - \int_0^\infty g(s) u_{xx}(t-s) {\rm d} s + u u_x = f $$ where the dissipation is entirely contributed by the memory term. Under a suitable smallness assumption on the external force $f$, we show that the related solution semigroup possesses the global attractor in the natural weak energy space. The result is obtained by means of a nonstandard approach based on the construction of a suitable family of attractors on certain invariant sets of the phase space.

For ν∈(0,1), we investigate the nonlinear integro‐differential equation on a multidimensional space domain Ω⊂RnDtνu−L1u−∫0tK(t−s)L2u(·,s)ds=f(x,t,u)+g(x,t)where Dtν is the Caputo fractional derivative and L1 and L2 are uniform elliptic operators with smooth coefficients depending on time. Under suitable conditions on the nonlinearity, the global existence and uniqueness of the classical solution to the related initial and boundary value problems are established.

We consider the MGT equation with memory $$\partial_{ttt} u + \alpha \partial_{tt} u - \beta \Delta \partial_{t} u - \gamma\Delta u + \int_{0}^{t}g(s) \Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $\alpha\beta>\gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g'$, namely, $$g'+\delta g\leq 0,\quad\delta>0,$$ but only asking that $g$ vanishes exponentially fast.

PurposeThe authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥q≥1.Design/methodology/approachCombinatorics, sine and cosine integral functions.FindingsThe authors prove an explicit formula to evaluate sinc-type integrals.Originality/valueThe proof is not present in the current literature, and it could be of interest for a large audience.

Given ρ ∈ [ 0 , 1 ] , we consider for e ∈ ( 0 , 1 ] the nonautonomous viscoelastic equation with a singularly oscillating external force ∂ t t u − κ ( 0 ) Δ u − ∫ 0 ∞ κ ′ ( s ) Δ u ( t − s ) d s + f ( u ) = g 0 ( t ) + e − ρ g 1 ( t / e ) together with the averaged equation ∂ t t u − κ ( 0 ) Δ u − ∫ 0 ∞ κ ′ ( s ) Δ 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 e ( t , τ ) acting on the natural weak energy space H are shown to possess uniform attractors A e . Within the further assumption ρ 1 , the family A e turns out to be bounded in H , uniformly with respect to e ∈ [ 0 , 1 ] . The convergence of the attractors A e to the attractor A 0 of the averaged equation as e → 0 is also established.

An abstract version of the fourth-order equation $$\partial_{tttt}u+\alpha\partial_{ttt}u+\beta\partial_{tt}u-\gamma\Delta\partial_{tt}u-\delta\Delta\partial_{t}u-\varrho\Delta u=0$$ subject to the homogeneous Dirichlet boundary condition is analyzed. Such a model encompasses the Moore–Gibson–Thompson equation with memory in presence of an exponential kernel. The stability properties of the related solution semigroup are investigated. In particular, a necessary and sufficient condition for exponential stability is established, in terms of the values of certain stability numbers depending on the strictly positive parameters $${\alpha, \beta, \gamma, \delta, \varrho}$$ .

We consider a linear Volterra integro-differential equation of hyperbolic type, which can be viewed as an abstract version of the equation $$\begin{aligned} \partial _{tt} u(t)- \varDelta u(t) +\displaystyle \int \limits _0^t\mu (s)\varDelta u(t-s)\mathrm{d}s=0 \end{aligned}$$describing the motion of linearly viscoelastic solids. We establish some decay results for the associated energy, under assumptions that do not involve differential inequalities for the convolution kernel $$\mu $$.

An elementary proof of a quantitative version of the Riemann-Lebesgue lemma for functions supported on the half line is given. Applications to differential models with memory are discussed.

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary modelin dimensionless form\begin{align*}\begin{cases}u'''' - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,d s\Big)u'' +k(u-v) = 0\\v'''' - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,d s\Big)v'' -k(u-v) = 0\end{cases}\end{align*}describing the equilibria of an elastically-coupled extensible double-beamsystem subject to evenly compressive axial loads.Necessary and sufficient conditionsin order to have nontrivial solutions are established, and their explicit closed-form expressions are found.In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes.In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutionsfor which the elastic energy fails to be evenly distributed.Such a feature turns out to be of some relevance in the analysis of the longterm dynamics,for it may lead up to nonsymmetricenergy exchanges between the two beams, mimickingthe transition from vertical to torsional oscillations.

We consider the model equation arising in the theory of viscoelasticity $$\partial_{tt} u-h_t(0)\Delta u -\int_{0}^\infty h_t'(s)\Delta u(t-s)d s+ f(u) = g.$$ Here, the main feature is that the memory kernel $h_t(\cdot)$ depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of Di Plinio et al. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.

We present a convergence–divergence test for series of nonnegative terms. Our proof is elementary, and yet we show examples of application to some apparently difficult cases.

For β ∈ ℝ, the authors consider the evolution system in the unknown variables u and α $$ \left\{ \begin{gathered} \partial _{tt} u + \partial _{xxxx} u + \partial _{xxt} \alpha - \left( {\beta + \left\| {\partial _x u} \right\|_{L^2 }^2 } \right)\partial _{xx} u = f, \hfill \\ \partial _{tt} \alpha - \partial _{xx} \alpha - \partial _{xxt} \alpha - \partial _{xxt} u = 0 \hfill \\ \end{gathered} \right. $$ describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.

In this survey paper, we discuss the decay properties of the semigroup generated by a linear integro-differential equation in a Hilbert space, which is an abstract version of the equation $${\partial_{tt}}u(t) - \Delta u(t) + {\int_0^\infty} \mu(s) \Delta u(t - s) {\rm{d}}s = 0$$ describing the dynamics of linearly viscoelastic bodies.

This note is concerned with the linear Volterra equation of hyperbolic type $$\partial _{tt} u(t) - \alpha \Delta u(t) + \int_0^t {\mu (s)\Delta u(t - s)} ds = 0$$ on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.

We consider differential systems with memory terms, expressed by convolution integrals, which account for the past history of one or more variables. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we focus on the reaction-diffusion equation with memory, and we discuss the convergence of solutions on finite time-intervals. When enough dissipativity is present, we also establish convergence results of the global and the exponential attractors. Nonetheless, the techniques here devised are quite general, and suitable to be applied to a large variety of models.

We consider a reaction-diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions. Copyright © 2005 John Wiley & Sons, Ltd.

We consider a conserved phase-field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth-order evolution equation for the phase-field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.

We consider a phase-field system with memory effects. This model consists of an integrodifferential equation of parabolic type describing the evolution of the (relative) temperature 9, and depending on its past history. This equation is nonlinearly coupled through a function A with a semilinear parabolic equation governing the order parameter X. The state variables 9 and X are subject to Neumann homogeneous boundary conditions. The model becomes an infinite-dimensional dynamical system in a suitable phase-space by introducing an additional variable η accounting for the (integrated) past history of the temperature. The evolution of η is thus ruled by a first-order hyperbolic equation. Giorgi, Grasselli, and Pata proved that the obtained dynamical system possesses a universal attractor A, which has finite fractal dimension provided that the coupling function A is linear. Here we prove, as main result, the existence of an exponential attractor E which entails, in particular, that A has finite fractal dimension when A is nonlinear with quadratic growth. Since the so-called squeezing property does not work in our framework, we cannot use the standard technique to construct E. Instead, we take advantage of a recent result due to Efendiev, Miranville, and Zelik. The present paper contains, to the best of our knowledge, the first example of exponential attractor for an infinite-dimensional dynamical system with memory effects. Also, the approach introduced here can be adapted to other dynamical systems with similar features.

We consider the coupled linear system $$\left\{\begin{array}{ll} \partial_{tt} u + \partial_{xxxx}u + \gamma \partial_t u + k(u - v) + h\partial_{t} (u-v) = 0\\\epsilon \partial_{tt} v - \partial_{xx}v - k(u - v) - h\partial_{t} (u - v) = 0\end{array}\right.$$ describing the vibrations of a string-beam system related to the well-known Lazer–McKenna suspension bridge model. For e > 0 and k > 0, the decay properties of the solution semigroup are discussed in dependence of the nonnegative parameters γ and h, which are responsible for the damping effects.

Every subalgebra of [Lscr ]([Hscr ]) which is isometrically weak*-homeomorphic with H ∞ (Ω) is reflexive, when Ω is a finitely connected region in the complex plane.

Hinčin characterized the class of infinitely divisible distributions on the line as the class of all distributional limits of sums of infinitesimal independent random variables. We show that an analogue of this characterization is true in the addition theory of free random variables introduced by Voiculescu.

The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.

For a family of semigroups Sε(t) : ℌε → ℌε depending on a perturbation parameter ε ∈ [0, 1], where the perturbation is allowed to become singular at ε = 0, we establish a general theorem on the existence of exponential attractors εε satisfying a suitable Hölder continuity property with respect to the symmetric Hausdorff distance at every ε ∈ [0, 1]. The result is applied to the abstract evolution equations with memorywhere kε(s) = (1/ε)k(s/ε) is the rescaling of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equationformally obtained in the singular limit ε → 0.

A classical result on the existence of global attractors for gradient systems is extended to the case of a semigroup S(t) lacking strong continuity, but satisfying the weaker property of being a closed map for every fixed t ≥ 0.

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernel which entails an extremely weak dissipation. In spite of that, we show that the related dynamical system possesses a global attractor of optimal regularity.

We consider, for ρ∈[0,1] and ε>0 small, the nonautonomous weakly damped wave equation with a singularly oscillating external force∂t2u−Δu+γ∂tu=−f(u)+g0(t)+ε−ρg1(t/ε), together with the averaged equation∂t2u−Δu+γ∂tu=−f(u)+g0(t). Under suitable assumptions on the nonlinearity and the external force, we prove the uniform (with respect to ε) boundedness of the attractors Aε in the weak energy space. If ρ

A semilinear strongly damped wave equation with memory is considered in the past history framework. The existence of global attractors of optimal regularity is established, both for critical and supercritical nonlinearities, under a necessary and sufficient condition on the memory kernel.

In this article, we study the long time behavior of a parabolic-hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen–Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double-well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient e is small enough. Then, we construct a robust family of exponential attractors (as e goes to 0). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

SUMMARY A weakly damped wave equation in the three-dimensional (3-D) space with a damping coecient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possessesnite-dimensional global and exponential attractors in a slightly weaker topology. Copyright ? 2006 John Wiley & Sons, Ltd.

Given a dissipative strongly continuous semigroup depending on some parameters, we construct a family of exponential attractors which is robust, in the sense of the symmetric Hausdorff distance, with respect to (even singular) perturbations.

We consider a hyperbolic phase-field model with memory which was introduced, justified, and studied in a previous authors'paper, jointly with C. Giorgi. There, we showed in particular that the model defines a dynamical system which possesses a uniformly absorbing set. Here, we deepen the analysis by proving that the system has a uniform attractor of finite fractal dimension, which is upper semicontinuous at zero with respect to a family of attractors associated with a class of parabolic phase-field systems with memory.

Equations with infinite delay commonly face the philosophical objection of being "unphysical", since a memory of infinite duration conflicts with reality. Indeed, besides common sense, experimental observations on concrete physical models tell that effects from the far past cannot possibly influence the current dynamics of a given system. On the other hand, infinite delay arises quite naturally in the mathematical description of several relevant phenomena. In this note, we propose a possible conceptual solution, showing that infinite delay can be recovered as a limiting case of finite delay on a large time-scale, along with a quantitative control of the discrepancy.

We consider a one-dimensional weakly damped wave equation, with a damping coefficient depending on the displacement. We prove the existence of a regular connected global attractor of finite fractal dimension for the associated dynamical system, as well as the existence of an exponential attractor.

In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore–Gibson–Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved.