A Nonlinear Hyperbolic Volterra Equation in Viscoelasticity.

Abstract

A general model for the nonlinear motion of a one dimensional, finite, homogeneous, viscoelastic body is developed and analysed by an energy method. It is shown that under physically reasonable conditions the nonlinear boundary, initial value problem has a unique, smooth solution (global in time), provided the given data are sufficiently "small" and smooth; moreover, the solution and its derivatives of first and second order decay to zero as t + -. Various modifications and generalizations, including two and three dimensional problems, are also discussed.

High-order essentially non-oscillatory scheme for viscoelasticity with fading memory

“…For the special kernel a of the form (1.5), which is the only case we consider in this paper, by defining Zj(x,t) = f CjHje~'1^t~T^ip(u(x,T))xdT, (2)(3)(4)(5)(6)(7)(8)(9)(10) Jo the equations (1.1)-(1.3) can be rewritten as…”

“…If the initial data Uq and vq and the body force g are smooth and small, the dissipation is strong enough to prevent the breakdown of smooth solutions. It has been shown that in this case the initial value problem (1.1)-(1.3) has a unique globally defined classical solution, which decays to the equilibrium as t -> oo; see [6], [3], and [20]; also see [12] and [4] for the initial boundary value problems. Large-time behavior of the solution has been studied in [21].…”

Global smooth solutions for a class of parabolic integrodifferential equations

“…If the kernel a is suitably decreasing and smooth up to t = 0, then local (in time) smooth solutions of (1.1a,b,c) can be found just as in the corresponding hyperbolic case with a = 0. For small initial data, these solutions exist for all times and remain small ( [4]), while for large data singularities as in the case with a = 0 may form in finite time ( [5], [13], [19]). Global existence results for smooth small solutions or more generally for small perturbations of arbitrary stationary solutions have been given in the case a(0) < ∞ = −a (0) in [21] and [26] and in the case a(0) = ∞ in [22], [29] and [32].…”

Non-linear PI regulators in control problems for holonomic mechanical systems

“…Using the representation of integral terms in the structure of control signals as regulators with unlimited after-effect (Anan'evskii & Kolmanovskii, 1989;Andreev, 2009), the motion of CONTACT Olga Peregudova peregudovaoa@gmail.com mechanical systems with these types of regulators can be modeled by Volterra integro-differential equations (Volterra, 1959). Such equations arise in the mathematical modelling of viscoelastic materials (Dafermos & Nohel 1981;Mac 1977;Sergeev 2007b;Volterra, 1959), population dynamics (Britton 1990;Volterra 1959), agedependent epidemic of a disease (El-Doma, 1987), nuclear reactor dynamics (Kappel & Di, 1972). The study of the qualitative theory of Volterra integro-differential equations including the stability problem attracts great attention of numerous researches, see for instance Burton (1983), Grimmer and Seifert (1975) and Sergeev (2007aSergeev ( , 2017 and their bibliographies.…”