Abstract
This paper considers the fuzzy viscoelastic model with a nonlinear source u t t + L u + ∫ 0 t g t − ζ Δ u ζ d ζ − u γ u − η Δ u t = 0 in a bounded field Ω. Under weak assumptions of the function g t , with the aid of Mathematica software, the computational technique is used to construct the auxiliary functionals and precise priori estimates. As time goes to infinity, we prove that the solution is global and energy decays to zero in two different ways: the exponential form and the polynomial form.
Highlights
In this paper, we take the following fuzzy viscoelastic model into account: ⎧⎪⎪⎪⎪⎨ utt + Lu t g(t − ζ)Δu(ζ)dζ − |u|cu − ηΔut 0
Journal of Mathematics e solution is perfectly stabilized through the dissipation, which is induced by the viscoelastic term. e modified energy functional in [10] has been used to prove the energy decay through two different ways: the exponential form and the polynomial form
We consider (1) in this paper, the two optimal decay rates, exponential decay and polynomial decay, are and directly established through the application of Mathematica software. e specific arrangement of this work is as follows: in Section 2, we present some notations and necessary materials; in Section 3, in view of the fuzzy number ƞ, we give the whole decay result, and our choice of the “Lyapunov” functional shows the extensive applicability and practical significance of the computational technique
Summary
We take the following fuzzy viscoelastic model into account:. ⎪⎪⎪⎪⎩ u(x, t) 0, u(x, t)|t 0 u0(x), ut(x, t)|t 0 u1(x), x ∈ Ω, t ∈ (0, ∞),. Journal of Mathematics e solution is perfectly stabilized through the dissipation, which is induced by the viscoelastic term. The authors in [11, 12] discussed the adaptive fuzzy control of nonlinear systems, and the discussion of fuzzy coefficients involved in these papers is quite interesting Inspired by these works, we consider (1) in this paper, the two optimal decay rates, exponential decay and polynomial decay, are and directly established through the application of Mathematica software. E specific arrangement of this work is as follows: in Section 2, we present some notations and necessary materials; in Section 3, in view of the fuzzy number ƞ, we give the whole decay result, and our choice of the “Lyapunov” functional shows the extensive applicability and practical significance of the computational technique We consider (1) in this paper, the two optimal decay rates, exponential decay and polynomial decay, are and directly established through the application of Mathematica software. e specific arrangement of this work is as follows: in Section 2, we present some notations and necessary materials; in Section 3, in view of the fuzzy number ƞ, we give the whole decay result, and our choice of the “Lyapunov” functional shows the extensive applicability and practical significance of the computational technique
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