Decay Estimates for a Type of Fuzzy Viscoelastic Integro-Differential Model

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.


Introduction
In this work, the following fuzzy viscoelastic integro-differential model is considered in a real Hilbert space X � L 2 (Ω): where Ω is an open bounded neighbourhood in R N with N ≥ 3, 0 < c ≤ (2/N − 2). Meanwhile, Γ: � zΩ is smooth enough. e memory kernel g(t) and the fuzzy number η are both positive, and g(t) is locally and absolutely continuous.
As far as the viscoelastic equation is concerned, profound research works have been made in many literature studies [1][2][3][4][5][6][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: e 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. e 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 integrodifferential abstract equation into account led to fruitful excellent results (see, e.g., [12][13][14]). In [12], Boussouira et al. proposed a unified method creatively. ey 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. e decay rates in exponential and polynomial forms, respectively, are straightly derived through the unified method. e 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 t ⟶ ∞, and the priori estimates are deduced without constructing any auxiliary function. Such outcomes reflect the reliability and effectiveness of the unified method in practice.

Preliminaries
roughout this work, the inner product 〈·, ·〉 of X will be utilized in its usual sense, and the norm is defined as follows: Note that Taking the operator into consideration, we can verify that where D(− Δ) is a dense domain. It is evident that, for some positive constant M, the linear operator − Δ is self-adjoint on the real Hilbert space L 2 (Ω) and satisfies an inequality similar to the Poincaré inequality [15]: What is more, we find that − Δ is accretive due to 〈− Δu, u〉 ≥ 0. Now, we give the following assumptions and preliminary materials about the memory kernel g(t).

Lemma 2. Consider a nonnegative nonincreasing function
then Complexity 3 Proof. Let en its derivative is calculated as Considering that is implies that On the other hand, since E is nonnegative and nonincreasing, Combining (30) with (31), we get Taking t � x + T, by x ≥ 0, formula (26) is obtained naturally.

Lemma 3. Let E(t) be a nonnegative and nonincreasing function on
where m, C, and T 0 are all positive constants. en, for arbitrary t ∈ [0, ∞), it holds that e proof of Lemma 3 is analogous to that of Lemma 2, and hence, it is omitted here.
Let u i ∈ X(i � 0, 1). Now, let us discuss the problem as follows: For any 0 ≤ t ≤ T, (T > 0), with the aid of the description in [12], a mild solution of (35) can be described as follows: and S(t) { } is the resolvent for the corresponding linear problem of (35).
As far as the weak solution is concerned, u is a function Local existence, uniqueness, and regularity for (1) are naturally guaranteed by the result in [12].
Considering a mild solution u of (1) (t ∈ [0, T]), and using u t as a multiplier, the multiplier method can be used to get the energy of u as follows: Next, it is necessary to discuss the decay of E u (t).
Consider that u is a strong solution of problem (1) on an interval [0, T]. By taking derivative of (39), we obtain 4 Complexity In view of the facts that g ≤ 0 and g ′ ≥ 0, it follows from these assumptions that that is, E u (t) 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 H 1 holds. For any u 0 ∈ D(∇) and u 1 ∈ X, if there is a positive scalar ρ 0 such that then there is a unique mild solution u for problem (1). Besides, for arbitrary t ∈ [0, ∞), Furthermore, u is a strong solution of (1), provided that u 0 ∈ D(− Δ) and u 1 ∈ D(∇).
Proof. Assume that a maximal definition interval for the mild solution of problem (1) If where ℓ ≥ ℓ + (η 2 /M). us, it is naturally acquired that Consequently,

Complexity
Utilizing the amplification method, i.e., So, the energy E u is well bounded and the solution u exists globally. e proof of the aforementioned formula is based on the idea of reductio ad absurdum, and the detailed process is omitted here.

Main Results
In this sequel, without invoking any auxiliary function, we put forward the main result as follows.
being a positive constant, then there is some positive constant C ensuring that the mild solution of model (1) satisfies the next property: Specifically, Proof. By eorem 1, we know that the solution of (1) is global. Moreover, it is easy to check that the solution is strong if u 0 ∈ D(− Δ) and u 1 ∈ D(∇). Aiming to show (53), we begin to focus on the formula as follows Next, our task is introducing an approach for controlling every term of the right hand of equation (55) via multiplier methods. At the beginning, we propose the following lemma. □ Lemma 4. Suppose that φ(t): R + ⟶ R + is a multiplier, fulfilling that φ ′ (t) < 0. en for any positive constant T with T ≥ S ≥ S 0 , there exists C > 0 such that Proof. Firstly, an inner product of model (1) with the multiplication of u and φ(t) should be taken. Next, integrating it on the closed interval [S, T], the following description is now obtained:

Complexity
Integrating by parts, we get Applying Schwartz inequality ∀ε 1 > 0, we have Taking the integrability of g ϑ and the assumption that g ′ ≤ − kg (p+1)/p into consideration, with the help of Hölder inequality and the description of E u ′ (t), we have Combining (62) and (63), we get In view of (45), we have us, we obtain erefore, the following result is arrived: (68) 8 Complexity erefore, By the condition imposed on the proof of eorem 1, one has Considering that both φ(t) and E(t) are decreasing, from (67), we deduce Based on equation (61), it is trivially shown that Complexity 9 which means that For simplicity, selecting where C 1 � 4 and are both positive. Next, multiplying both sides of the original equation (1) by φ(t) at the same time, taking t 0 g(t − ζ)(u(ζ) − u(t))dζ as a multiplier, and integrating on the closed interval [S, T], the following equation can be obtained: Taking integration by parts, one obtains 10 Complexity Substituting (79) and (80) into (78) leads to the following: Now, let us consider the term T S φ(t)‖u t ‖ 2 dt and evaluate it. First of all, according to (10), we have

Complexity 13
For any S 0 ∈ (0, S], with the help of the fact g(0) > 0 and the continuity of g, it can be acquired that Choosing a positive constant δ 3 which is small enough so that δ 3 < S 0 0 g(ζ)dζ, and considering we can check that Now, for any S ∈ [S 0 , T), we have us, we can conclude that where ε 4 � (2ε 3 /δ 3 ), and Consequently, If ε 5 is small enough in the above formula, then Alternatively, if C 3 is taken properly, the estimation can be arrived as Considering the third term and the fourth one of (56), we get By combining equations (100)-(102), it is shown that (56) is true, which concludes the proof.
Next, it remains to complete the Proof of eorem 2.

□
Proof of eorem 2. Let us consider the case where p equals infinity firstly. For any t, if c represents any positive constant, then by taking φ(t) � c in (56), we can get the following result:

Complexity
Again, g(t) ≤ − (g ′ (t)/k) follows from g ′ (t) ≤ − kg(t). Invoking Lemma 3, one may deduce that en, is fact further explains the attenuation of E u (t) according to a polynomial form.
Secondly, it is valuable to consider the case of 2 < p < ∞. Aiming to evaluate the last term of E u (t), we will put forward the following lemmas.
As T tends to infinity, the limit result of long-time memory is easily seen, and thus, (114) is true. □ Remark 3. With the aid of paper [12], it is trivial to show that ∞ S ε (p+1)/p u (t)dt ≤ C 8′ ε u (S) ε (1/p) e proof of (120) is entirely similar to that of (114) and so it is omitted here. Now, let us turn back to complete the verification of eorem 2.

Conclusion
Based on the proposed appropriate assumptions of the convolution kernels along with the discussion about the fuzzy number η, the exponential and polynomial aspects of the energy decay rates for system (1) are estimated only through the application of the multiplier method and the unified technique. In this process, the most valuable point is that our research has avoided the construction of auxiliary functions perfectly. e appearance of the term with fuzzy coefficient makes the expression form of E u richer and it leads to some difficulties in calculation. At the same time, more efforts have been spent on discussing the integro-differential inequalities and the discussion is quite interesting. Considering the case of η � 0, we can see that the results coincide with that of reference [12]. In summary, the result in this paper reveals the wide applicability of the unified method, and further discussion for the blow-up problems may be considered in the future.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.