On the Properties of Semigroups Generated by Volterra Integro-Differential Equations
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On the Properties of Semigroups Generated by Volterra Integro-Differential Equations V. V. Vlasov1,2∗ and 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], [email protected], ∗∗ [email protected] Received April 28, 2020; revised April 28, 2020; accepted May 14, 2020
Abstract— We study abstract integro-differential equations serving as operator models for problems arising in viscoelasticity. The results are based on an approach related to the study of one-parameter semigroups for linear evolution equations. The approach proposed can be used when investigating other integro-differential equations containing integral terms of the form of Volterra convolution. DOI: 10.1134/S001226612008011X
INTRODUCTION The paper presents results on the existence of a strongly continuous contraction semigroup generated by a Volterra integro-differential equation with operator coefficients in a Hilbert space. The exponential decay of this semigroup is established under some assumptions about the kernels of the integral operators. Based on these results, the well-posed solvability of the original initial value problem is established for the Volterra integro-differential equation and appropriate estimates are obtained for the solution. We give examples of exponential and fractional-exponential integral operator kernels (Rabotnov’s functions) satisfying our assumptions. 1. DEFINITIONS. NOTATION. STATEMENT OF THE PROBLEM Let H be a separable Hilbert space, and let A = A∗ ≥ κ0 I (κ0 > 0) be a positive self-adjoint operator with bounded inverse in H. Let B be a symmetric operator, (Bx, y) = (x, By), with domain Dom (B) (Dom (A) ⊆ Dom (B)) in H; we assume that B is nonnegative, (Bx, x) ≥ 0 for each x ∈ Dom (B), and satisfies the inequality kBxk ≤ κkAxk, 0 < κ < 1, for all x ∈ Dom (A); I is the identity operator in H. Consider the following problem for a second-order integro-differential equation on the positive half-line R+ ≡ (0, ∞): t
m Z X d2 u(t) + (A + B)u(t) − Rk (t − s)(ak A + bk B)u(s) ds = f (t), dt2 k=1
t ∈ R+ ,
(1)
0
u(+0) = ϕ0 ,
u(1) (+0) = ϕ1 ,
(2)
with coefficients ak > 0, bk ≥ 0, k = 1, . . . , m. Assume that the functions Rk : R+ → R+ satisfy the following conditions: Rk (t) are positive nonincreasing functions, Rk (t) ∈ L1 (R+ ), lim Rk (t) = 0, k = 1, . . . , m. t→+∞
Set
Z+∞ Z+∞ Mk (t) = Rk (s) ds = Rk (t + s) ds, t
0
1100
k = 1, . . . , m.
(3)
ON THE PROPERTIES OF SEMIGROUPS
1101
In addition, we assume that the following inequalities hold: m X
m X (bk Mk (0)) < 1.
(ak Mk (0)) < 1,
k=1
Let A0 =
1−
m X
(4)
k=1
! 1−
ak Mk (0) A +
k=1
m X
! bk Mk (0) B,
Ak = ak A + bk B.
k=1
It follows from a well-known result (see [1, p. 361 of the Russian translation]) that the operators A0 and Ak , k = 1, . . . , m, are self-adjoint and positive. Note that problems of the form (1), (2) are operator models of problems arising in viscoelasticity (see [2, 3
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