Abstract

We consider a type of fuzzy viscoelastic integro-differential model in this paper. With the aid of some appropriate hypotheses, a unified method and the multiplier technique are implemented to get priori estimates precisely without constructing any auxiliary function. By establishing the estimation of energy function, we derive the stability result of the global solution, and we calculate the estimations of energy attenuation in exponential and polynomial forms, respectively.

1. Introduction

In this work, the following fuzzy viscoelastic integro-differential model is considered in a real Hilbert space :where is an open bounded neighbourhood in with , . Meanwhile, is smooth enough. The memory kernel and the fuzzy number are both positive, and is locally and absolutely continuous.

As far as the viscoelastic equation is concerned, profound research works have been made in many literature studies [1–7]. For example, the authors in [3] proved a local existence theorem for the next equation:which is subject to some proper initial data and conditions. In [4], an appropriate Lyapunov-type function was introduced by Nasser-eddine Tatar to prove the decay of solutions for the wave equation:

The key contribution of ref. [6] is that the authors demonstrated the decay of the energy function for the next wave equation:and some Lyapunov functions were exploited felicitously to deduct more general energy decay results. In [8], a nonlinear hereditary memory evolution equation was considered, and several stability results were given just by means of a simple auxiliary function. The authors in [9] attained analytical and approximate solutions for the cubic Boussinesq equations and modified ones with the aid of the He–Laplace method. Besides, fuzzy synchronization problems have captured the intensive interests of scholars (see, e.g., [10, 11]), where an adaptive fuzzy backstepping control method was developed in ref. [11] for a sort of uncertain fractional-order nonlinear system.

Generally speaking, in most of the existing works, the presence of auxiliary functions is inevitable, which is exploited to seek the attenuation result of the solution. Accordingly, in the discussion of energy attenuation of solutions for the fuzzy viscoelastic integro-differential model, how to reduce the construction of auxiliary functions has become a problem worth discussing. Taking the integro-differential abstract equation into account led to fruitful excellent results (see, e.g., [12–14]). In [12], Boussouira et al. proposed a unified method creatively. They derived the decay results for second-order integro-differential equations in the following abstract form:

Also, they put forward an exquisite unified method. With the help of the multiplier method, they accurately described the energy attenuation of the solution of the abstract equation mentioned above.

Inspired by these works, system (1) involved in this paper is an extension of the equation appeared in [12], in which a term with fuzzy coefficient is creatively added. The decay rates in exponential and polynomial forms, respectively, are straightly derived through the unified method. The specific arrangement is made as follows: firstly, in Section 2, several preliminary materials and essential assumptions are listed, and secondly, Section 3 mainly concentrates on the global solution and the estimation of energy attenuation, which are derived by letting , and the priori estimates are deduced without constructing any auxiliary function. Such outcomes reflect the reliability and effectiveness of the unified method in practice.

2. Preliminaries

Throughout this work, the inner product of will be utilized in its usual sense, and the norm is defined as follows:

Note that

Taking the operatorinto consideration, we can verify thatwhere is a dense domain. It is evident that, for some positive constant , the linear operator is self-adjoint on the real Hilbert space and satisfies an inequality similar to the Poincaré inequality [15]:

What is more, we find that is accretive due to .

Now, we give the following assumptions and preliminary materials about the memory kernel .

: as far as is concerned, for some and , the function fulfills the following conditions:

Remark 1. If , then yields with , which indicates that will decay exponentially.
If , then yieldsi.e.,Next, based on the aforementioned results, the following expression can be obtained by integrating from 0 to :namely,This means that will decay polynomially. Simultaneously, by generalized integral property, if , it may imply that .

Lemma 1. Suppose that

The function is Gateaux differentiable for every , and

Indeed, it is straightforward to see that . For any pair , there exists such that

in which .

For all with , where , it is easily seen that . Combining the mean value formula applied in [12] and (10), one can verify that

That is, some positive constant can be found to satisfy

By letting , it is easy to see that is continuous and strictly increasing. Now, we suppose that , that is,

For every ,which yields

Remark 2. For any , by taking the measurable function into consideration, it is known that both and are finite.
For any and , we denote the convolution as follows:Aiming to facilitate the subsequent narration, we proceed to present the next useful lemmas.

Lemma 2. Consider a nonnegative nonincreasing function with . If there exists a negative constant such thatthen

Proof. LetThen its derivative is calculated asConsidering that , we haveThis implies thatOn the other hand, since is nonnegative and nonincreasing,Combining (30) with (31), we getTaking , by , formula (26) is obtained naturally.

Lemma 3. Let be a nonnegative and nonincreasing function on . Ifwhere , and are all positive constants. Then, for arbitrary , it holds that

The proof of Lemma 3 is analogous to that of Lemma 2, and hence, it is omitted here.

Let . Now, let us discuss the problem as follows:

For any , with the aid of the description in [12], a mild solution of (35) can be described as follows:whereand is the resolvent for the corresponding linear problem of (35).

As far as the weak solution is concerned, is a function in and satisfies and .

Local existence, uniqueness, and regularity for (1) are naturally guaranteed by the result in [12].

Considering a mild solution of (1) , and using as a multiplier, the multiplier method can be used to get the energy of as follows:

Next, it is necessary to discuss the decay of .

Consider that is a strong solution of problem (1) on an interval . By taking derivative of (39), we obtain

In view of the facts that and , it follows from these assumptions thatthat is, is decreasing. One can draw a similar conclusion for mild solutions. In a word, if the initial conditions are small sufficiently, the solution of model (1) exists globally.

Theorem 1. Assume that holds. For any and , if there is a positive scalar such thatthen there is a unique mild solution for problem (1). Besides, for arbitrary ,

Furthermore, is a strong solution of (1), provided that and .

Proof. Assume that a maximal definition interval for the mild solution of problem (1) is , and . According to Lemma 1, one gets . Besides, equation (39) impliesIf , we getwhere .
Thus, it is naturally acquired thatConsequently, .
Put . Suppose that and satisfyUtilizing the amplification method, i.e.,we derive that andSo, the energy is well bounded and the solution exists globally. The proof of the aforementioned formula is based on the idea of reductio ad absurdum, and the detailed process is omitted here.