Semigroups Generated by Volterra Integro-Differential Equations
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GRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
Semigroups Generated by Volterra Integro-Differential Equations N. A. Rautian1,2∗ 1
2
Lomonosov Moscow State University, Moscow, 119991 Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia e-mail: ∗ [email protected] Received May 12, 2020; revised June 11, 2020; accepted June 26, 2020
Abstract—We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of oneparameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation. The existence of a contraction C0 -semigroup is proved under certain assumptions about the kernels of integral operators. Examples of exponential and fractional-exponential (Rabotnov functions) kernels of integral operators satisfying the above assumptions are given. DOI: 10.1134/S0012266120090098
INTRODUCTION The origins of the modern concept of viscoelasticity and, in a more general sense, the so-called hereditary systems originate in the works of Boltzmann and Volterra, who introduced the concept of memory in connection with the analysis of viscoelastic materials. The results of numerous experiments show that the constitutive equations for a majority of existing materials can be assumed to be linear within the limits of small deformations. Next, we will consider the abstract integro-differential equation arising in the theory of linear viscoelasticity and present a general scheme for studying this equation. This scheme can also be applied to many other linear models containing Volterra integral operators. This abstract integro-differential equation can be realized as an integro–partial differential equation of the form utt (x, t) = ρ−1 µ∆u(x, t) + 3−1 (µ + λ) grad (div u(x, t)) Zt − K(t − τ )ρ−1 µ ∆u(x, τ ) + 3−1 grad (div u(x, τ )) dτ (1) 0 t Z − Q(t − τ ) 3−1 ρ−1 λ grad (div u(x, τ )) dτ + f (x, t), 0 3
where u = ~u(x, t) ∈ R is the vector of small displacements of a viscoelastic isotropic medium filling a bounded domain Ω ⊂ R3 with smooth boundary ∂Ω, the density ρ > 0 is constant, and λ and µ are positive parameters (Lam´e coefficients) (see [1–3]). We will assume that the Dirichlet conditions u|∂Ω = 0 are satisfied on the boundary of the domain Ω. The kernel functions K(t) and Q(t) of the integral operators are positive nonincreasing integrable functions characterizing the hereditary properties of the medium. The analysis of the integro-differential equation (1) becomes much simpler if we assume that the kernels K(t) and Q(t) of the integral operators are sums of decaying exponentials with positive coefficients; in particular, K(t) =
m X
ck e−βk t ,
t ≥ 0,
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