Uniform Energy Decay Rates for the Fuzzy Viscoelastic Model with a Nonlinear Source

Abstract

This paper considers the fuzzy viscoelastic model with a nonlinear source in a bounded field Ω. Under weak assumptions of the function , 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.

1. Introduction

In this paper, we take the following fuzzy viscoelastic model into account:where the fuzzy number

In , Ω is a domain which is well bounded. Besides, the boundary of Ω is smooth perfectly and expressed as . Meanwhile, the memory kernel (t) is positive and some assumptions will be given in detail.

This type of problems has been observed in many areas of scientific and engineering fields. For example, time analyticity for the viscoelastic equation is studied as follows [1]:

S. Berrimi and S. A. Messaoudi applied weak conditions on the memory kernel . Meanwhile, considering the condition that the energy is positive and relatively small, they obtained the existence of global solutions.

Takinginto account, they also obtained a decay rate exponentially in [2]. M.M. Cavalcanti and H.P. Oquendo improved this latter result in [3]. In their work, two situations, the internal dissipation and the viscoelastic dissipation, are considered to act on respective part separately. The authors in [3] expanded the internal dissipation to nonlinear cases as much as possible. Simultaneously, the system is well stabilized through the dissipation which is induced by the integral term.

As we know, the viscoelastic terms attract many mathematicians; for instance, the authors in [4] studied the energy decay rate for the global solution of a quasilinear viscoelastic model, and the authors in [5–9] considered time analysis of solutions for some viscoelastic models. Moreover, F. Y. Zhang et al. considered a nonlinear viscoelastic equation in [10]:

The solution is perfectly stabilized through the dissipation, which is induced by the viscoelastic term. The modified energy functional in [10] has been used to prove the energy decay through two different ways: the exponential form and the polynomial form. Additionally, the construction of auxiliary functions is organized by the computational technique of undetermined coefficients. Besides, 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 easily and directly established through the application of Mathematica software. The 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.

2. Preliminaries

In this section, and are understood and applied in their usual senses. We impose the following hypotheses and preliminaries on the memory kernel (t). In addition, the definition of energy function plays a significant role to our main result: H₁: as a bounded function, satisfies where both (0) and are positive. H₂: the existence of a positive constant makes the following formula hold:

Remark 1. From the assumption above, if , we haveIf , we havewhere .
Indeed, the condition plays an important role to ensure that

Theorem 1. Assuming that , then u (t) can be found as a unique solution to model (1) with

Simultaneously, we get

Taking the initial data, less regularity, and the priori estimate into consideration, the theorem above guarantees the existence of solutions for model (1) as in [7, 13]. In addition, the Galerkin approximation method can be applied to accomplish the proof of the theorem above.

Our primary assignment is to find out the energy function . Combining the multiplier method, the integral subsection integration, and (H₁, H₂), the calculation is provided:

Now, (13) yields

Now, it is obvious that the energy of model (1) can be given directly:where

Thereby, we can findwhich means is decreasing.

3. Main Results

In this chapter, the main result of this work is put forward directly and clearly. We introduce the following auxiliary functionals firstly:where the coefficients will be determined later.

Lemma 1. The existence of a positive constant C makes the following conclusion:

Proof. Recalling the inequalities of Cauchy and Poincaré, the following estimates can be arrived:where the symbol satisfies for all .
Here, it is necessary to indicate that the positive constants later in this article represent different meanings in different places.
From the representation of , it is naturally for us to getThus, (20) follows.

Lemma 2. For any , the following estimate holds:where are selected appropriately.

Proof. Deriving function with respect to time, the following calculation will be given clearly:Considering the fourth item of (23), we adopt the following estimate from [10] without proof:Taking the sixth item of (23) into account, we adopt the following estimate from [1] as follows:For the last item of (23), we haveThus, (21) follows.

Lemma 3. For , the following estimate holds:where , are selected appropriately.

Proof. Deriving function with respect to time, the expression will be given distinctly:For any we will estimate every item of the right hand of (29):Similar to the first term, considering the third item of (29), similar estimate will be given:Applying Young’s inequality properly, the following estimates can be obtained from [10]:Similarly, we haveFurthermore,Taking (29)–(39) and H₂ into account, for all , we get (28).

Theorem 2. Let every pair , H₁ and H₂ hold, and For every , there must be some positive constants K₁, K₂, and k, which would enable the solution of the model (1) to satisfy the following:

Proof. We start the proof by selecting an appropriate auxiliary functionalwhere the positive constants M, a, and b will be chosen in the sequel.
Applying Lemma 1 and the hypothesis , the equivalence between and is achieved. To this point, firstly, a simple calculation shows thatBy (25), we getThus, the equivalence between and is complete.
Secondly, differentiating (41) with respect to t, considering the assumption H₂ and Lemmas 1 and 2, the estimate of is given naturally:where the coefficients are given as follows:Next, it is time to carry out a discussion on the different values of p.

Case 1.
In such a case, choosing and taking the positive constant h aswe findTherefore, (45) yields

Remark 2. Using the computational technique proposed in [10] and selecting some appropriate numerical values by means of the calculation software Mathematica, it is obvious that .
Considering the interval of the fuzzy number η, takingfor example, with the aid of Mathematica, we easily verify thatTaking another example into consideration,we find thatAdditionally, we may retrieve more effective data of , and the data provide an intuitive understanding in the discussion of the auxiliary functionals.
Considering the equivalence between and , it is easy for us to see thatwhere the coefficients M₁ and M₂ in (53) have many possibilities to be chosen.
The following expression can be presented through the combination of (48) and (53):After a simple integral on the interval , (54) leads toAs a consequence, taking , (55) leads towhere K₁ is some appropriate positive constant.

Case 2. .
Using H₁, H₂, and (9), we easily verify thatMoreover, considering the inequality in [6], for some constant C,Taking , we find Therefore, for some , we getthat is,Taking (48) into account, we haveFor each , it is straight for us to getAs we mentioned before, the symbol C denotes different constants in different places. Executing a simple integral on the interval , (62) yieldsConsidering the equivalence between and , it is apparently for us to getThis completes the proof.

4. Conclusion

Under the assumptions on the relaxation function and the interval of the fuzzy number η, applying the computational technique, a lot of auxiliary functionals can be constructed numerically. Two decay results, the exponential one and the polynomial one, are derived for the model (1) eventually. The result shows a new way for the decay rates, which is quite different from other literatures.

Data Availability

All datasets generated for this study are included in the manuscript.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by the Youth Foundation of Jining University (2019QNKJ03).