Abstract
In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and a relaxation function satisfying g′(t) ≤ −ξ(t)G(g(t)). Using the Galaerkin method, we establish the existence of the solution and prove an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This work generalizes and improves earlier results in the literature. In particular, those of Messaoudi (2016) and Mustafa (Math Methods Appl Sci. 2017;V41:192–204).
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1 Introduction
In this paper, we consider the following viscoelastic problem:
where u denotes the transverse displacement of waves and Ω is a bounded domain of \(\mathbb {R}^{N} (N\ge 1)\) with a smooth boundary ∂Ω, g is positive and decreasing function and m > 1.
The study of viscoelastic problems has attracted the attention of many authors and several decay and blow up results have been established. In [6], Cavalcanti et al. considered the equation
where \(a:{\Omega } \rightarrow \mathbb {R}^{+}\) is a function which may vanish on a part of the domain Ω but satisfies a(x) ≥ a0 on ω ⊂Ω and g satisfies, for two positive constants ξ1 and ξ2,
They established an exponential decay result under some restrictions on ω. Berrimi and Messaoudi [4] established the result of [6], under weaker conditions on both a and g, to a problem where a source term is competing with the damping term. Fabrizio and Polidoro [11] studied the following system
and showed that the exponential decay of the relaxation function is a necessary condition for the exponential decay of the solution energy. Cavalcanti and Oquendo [7] considered the following problem
and established, for a(x) + b(x) ≥ ρ > 0, an exponential stability result for g decaying exponentially and h linear and a polynomial stability result for g decaying polynomially and h nonlinear. Rivera [26] considered equations for linear isotropic homogeneous viscoelastic solids of integral type which occupy a bounded domain or the whole space \(\mathbb {R}^{n}\), with zero boundary and history data and in the absence of external body forces. In the bounded domain case, an exponential decay result was proved for exponentially decaying memory kernels and for the whole space case a polynomial decay result was established and the rate of the decay was given. This latter result was later pushed to a situation where the kernel is decaying algebraically but not exponentially by Cabanillas and Rivera [5]. In their paper, the authors showed that the decay of solutions is also algebraic, at a rate which can be determined by the rate of the decay of the relaxation function and may be improved by the regularity of the initial data. The authors considered both cases, the bounded domains and that of a material occupying the entire space. This result was later improved by Baretto et al. [3], where equations related for linear viscoelastic plates were treated. Precisely, they showed that the solution energy decays at the same decay rate of the relaxation function. For partially viscoelastic materials, Rivera et al. [27, 28] showed that solutions decay exponentially to zero, provided the relaxation function decays in a similar fashion, regardless to the size of the viscoelastic part of the material.
In 2008, Messaoudi [21, 22] generalized the decay rates allowing an extended class of relaxation functions and gave general decay rates from which the exponential and the polynomial decay rates are only special cases. However, the optimality in the polynomial decay case was not obtained. Precisely, he considered relaxation functions that satisfy
where \(\xi :\mathbb {R}^{+} \to \mathbb {R}^{+}\) is a nonincreasing differentiable function and showed that the rate of the decay of the energy is the same rate of decay of g, which is not necessarily of exponential or polynomial decay type. After that, a series of papers using Eq. 1.4 has appeared (see, for instance, [13, 19, 20, 25, 29, 30, 34, 35]).
Inspired by the experience with frictional damping initiated in the work of Lasiecka and Tataru [15], another step forward was done by considering relaxation functions satisfying
This condition, where χ is a positive function, χ(0) = χ′(0) = 0, and χ is strictly increasing and strictly convex near the origin, with some additional constraints imposed on χ, was used by several authors with different approaches. We refer to previous studies [1, 8, 9, 12, 16, 17, 31] and [36], where general decay results in terms of χ were obtained. Here, it should be mentioned that, in [17], it was the first time where Lasiecka and Wang established not only general but also optimal results in which the decay rates are characterized by an ODE of the same type as the one generated by the inequality (1.5) satisfied by g. Mustafa and Messaoudi [33] established an explicit and general decay rate for relaxation function satisfying
where \(H\in C^{1}(\mathbb {R})\), with H(0) = 0 and H is linear or strictly increasing and strictly convex function C2 near the origin. In [10], Cavalcanti et al. considered the following problem
with a relaxation function satisfying (1.6) and the additional requirement:
and that \(y^{1-\alpha _{0}}\in L^{1}(1,\infty )\), for some α0 ∈ [0, 1), where y(t) is the solution of the problem
They characterized the decay of the energy by the solution of a corresponding ODE as in [15]. Recently, Messaoudi and Al-Khulaifi [24] treated (1.7) with a relaxation function satisfying
They obtained a more general stability result for which the results of [21, 22] are only special cases. Moreover, the optimal decay rate for the polynomial case is achieved without any extra work and conditions as in [16] and [15]. Very recently, Mustafa [32] answered the question when he studied a viscoelastic equation with relaxation function satisfies (2.2) (below) and established an optimal decay result using the multiplier method and some properties of the convex functions. In this paper, we intend to extend the results of Messaoudi [23] and Mustafa [32] to Eq. 1.1.
This paper is organized as follows. In Section 2, we present some notations and material needed for our work. In Section 3, we establish the global existence of the solution of the problem. Some technical lemmas and the decay results are presented in Sections 4 and 5, respectively.
2 Preliminaries
In this section, we present some materials needed in the proof of our results. We use the standard Lebesgue space L2(Ω) and Sobolev space \({H_{0}^{1}}({\Omega })\) with their usual scalar products and norms. Throughout this paper, c and ε are used to denote generic positive constants.
We consider the following hypotheses:
(A1) \(g: \mathbb {R}^{+}\to \mathbb {R}^{+}\) is a C1 nonincreasing function satisfying
$$ g(0) > 0, \hspace{0.2in}\ 1-{\int}_{0}^{+\infty}g(s)ds={\ell} > 0, $$(2.1)and there exists a C1 function G : (0, ∞) → (0, ∞) which is linear or it is strictly increasing and strictly convex C2 function on (0, r], r ≤ g(0), with G(0) = G′(0) = 0, such that
$$ g^{\prime}(t)\le -\xi(t) G(g(t)),\hspace{0.15in}\forall t\ge 0, $$(2.2)where ξ(t) is a positive nonincreasing differentiable function.
(A2) For the nonlinearity in the damping, we assume that
$$\begin{array}{@{}rcl@{}} &&1<m\le\frac{2n}{n-2},\hspace{0.05in}\text{if}\hspace{0.05in}n>2\\ &&\text{and}\\ && m>1,\hspace{0.05in}\text{if}\hspace{0.03in}n = 1,2. \end{array} $$(2.3)
We introduce the “modified” energy associated to problem (1.1)
where
Direct differentiation, using Eq. 1.1, leads to
Remark 2.1
If G is a strictly increasing and strictly convex C2 function on (0, r], with G(0) = G′(0) = 0, then it has an extension \(\overline {G}\), which is strictly increasing and strictly convex C2 function on (0, ∞). For instance, if G(r) = a, G′(r) = b, G″(r) = c, we can define \(\overline {G}\), for t > r, by
3 Existence
In this section, we state and prove an existence result of problem (1.1).
Definition 3.1
For any pair \((u_{0},u_{1})\in {H^{1}_{0}}({\Omega })\times L^{2}({\Omega })\). A function
is called a weak solution of Eq. 1.1 if
Proposition 3.2
Let\((u_{0},u_{1})\in {H_{0}^{1}}({\Omega })\times L^{2}({\Omega })\)begiven. Assume that (A1) and (A2) hold. Then problem (1.1) has a unique weak global solution.
Proof
We use the standard Faedo-Galerkin method to prove our result. Let \(\{w_{j}\}_{j = 1}^{\infty }\) be the eigenfunctions of the Laplacian operator subject to Dirichlet boundary conditions. Then \(\{w_{j}\}_{j = 1}^{\infty }\) is orthogonal basis of \({H_{0}^{1}}({\Omega })\) as well as of L2(Ω). Let Vk = span{w1, w2, ... , wk} and the projections of and initial data on the finite-dimensional subspace Vk are given by
where,
We search solutions of the form
for the approximate problem in Vk
This leads to a system of ODE’s for unknown functions hj, k. Based on standard existence theory for ODE, the system (3.3) admits a solution uk on a maximal time interval [0, tk), 0 < tk < T, for each \(k\in \mathbb {N}.\) In fact tk = T = + ∞ and to show this, let \(w={u_{t}^{k}}\) in Eq. 3.3 and integrate by parts to obtain
where
Integrate (3.4) over (0, t) to obtain
This means, using (A1) and Eq. 3.2, that, for some positive constant C independent of t and k,
Thus, we can extend tk to infinity and, in addition, we have
Therefore, there exists a subsequence of (uk),still denoted by (uk),such that
Since \(({u_{t}^{k}})\) is bounded in Lm(Ω × (0, T)), then \(({\vert {u_{t}^{k}}\vert }^{m-2}{u_{t}^{k}})\) is bounded in \(L^{\frac {m}{m-1}}({\Omega }\times (0,T)).\) Hence, up to a subsequence,
Now, our task to show that ψ = |ut|m− 2ut. For this purpose, integrate (3.3) over (0, t) to obtain
Convergences (3.2), Eqs. 3.6 and 3.7 allow us to pass to the limit in Eq. 3.8, as k → + ∞, and get
which implies that Eq. 3.9 is valid for any \(w\in {H_{0}^{1}}({\Omega }).\) Using the fact that the left hand side of Eq. 3.9 is an absolutely continuous function, hence it is differentiable for a.e t ∈ (0, ∞), and we get
Now, define
This is true by the following elementary inequality (see Theorem 6.1, p. 222 [18]):
So, by using Eq. 3.5, we get
Taking k → + ∞, we obtain
Replacing w by ut in Eq. 3.10 and integrating over (0, T),we obtain
Combining Eqs. 3.13 and 3.14, we arrive at
Hence,
by density of \({H^{1}_{0}}({\Omega })\) in Lm(Ω). Let v = λz + ut, z ∈ Lm(Ω × (0, T)). So, we get, ∀λ ≠ 0,
Let λ > 0. So we have
As λ → 0,we get
Similarly, for λ < 0,we get
Thus, Eqs. 3.15 and 3.16 imply that ψ = |ut|m− 2ut. Hence Eq. 3.10 becomes
To handle the initial conditions, we note that
Thus, using Lion’s Lemma [18] and Eq. 3.2, we easily obtain
As in [14], multiply (3.3) by \(\phi \in C^{\infty }_{0}(0,T)\) and integrate over (0, T), we obtain for any w ∈ Vk
As k → + ∞, we have for any \(w\in {H^{1}_{0}}({\Omega })\) and any \(\phi \in C^{\infty }_{0}((0,T)),\)
This means (see [14]),
Recalling that ut ∈ L2((0, T), L2(Ω)), we obtain
So, \({u^{k}_{t}}(x,0)\) makes sense and
But
Hence
For uniqueness, let us assume that problem (1.1) has two solutions u and v. Then, w = u − v satisfies
Now, multiply (3.20) by wt and integrate over Ω × (0, t) to obtain
Hence, by using inequality (3.12), we have
which implies that w = C. In fact, C = 0 since w = 0 on ∂Ω. Which completes the proof. □
4 Technical Lemmas
In this section, we establish several lemmas needed for the proof of our main result. We adopt some results from [23] and [32] without proof.
Lemma 4.1
For\(u \in {H^{1}_{0}}\left ({\Omega }\right )\), we have
where, for any 0 < α < 1,
Proof
The Use of Eq. 4.2 and the Cauchy Schwarz inequality gives
□
Lemma 4.2
[23, 32] Under the assumptions (A1) and (A2), the functional
satisfies, along the solution, the estimate
and
Lemma 4.3
[23, 32] Under the assumptions (A1) and (A2), thefunctional
satisfies, along the solution, the estimate
and
Lemma 4.4
[32] Under the assumptions (A1) and (A2), the functional
satisfies, along the solution of Eq. 1.1, the estimate
where\(r(t)={\int }_{t}^{+\infty }g(s)ds.\)
Proof
By Young’s inequality and the fact that r′(t) = −g(t),we see that
Now,
Using the facts that r(t) ≤ r(0) = 1 − ℓ and \({{\int }_{0}^{t}}g(s)ds \le 1-\ell \), Eq. 4.9 is established. □
Lemma 4.5
[32] There exist positive constantsd andt1such that
Proof
By (A1),we easily deduce that \(\lim _{t\to +\infty }g(t)= 0.\) Hence, there is t1 > 0 large enough such that
and
As g and ξ are positive nonincreasing continuous and G is a positive continuous function, then, for all t ∈ [0, t1],
which implies that there are two positive constants a and b such that
Consequently, for all t ∈ [0, t1],
□
Remark 4.6
Using the fact that \(\frac {\alpha g^{2}(s)}{\alpha g(s)-g^{\prime }(s)} <g(s)\) and recalling the Lebesgue dominated convergence theorem, we can easily deduce that
Lemma 4.7
Assume that (A1) and (A2). Then there exist strictlypositive constantsN, ε1, ε2, λ, csuch that the functional
satisfies, for all t ≥ t1,
and
Proof
For the proof of Eq. 4.13, we refer the reader to [22]. Now, we prove inequality (4.14). Let \(g_{1}:={\int }_{0}^{t_{1}}g(s)ds>0\). By using Eqs. 2.5, 4.4 and 4.6, recalling that g′ = (αg − h) and taking \(\delta =\frac {\ell }{4N_{2}}\), we easily see that, for all t ≥ t1,
At this point, we choose N1 large enough so that
and then N2 large enough so that
Now, using Remark 4.6, there is 0 < α0 < 1 such that if α < α0, then
Next, we choose N large enough so that
which gives
Therefore, we arrive at
Combining Eqs. 2.4 and 4.18, Eq. 4.14 is established. The same calculations hold, for m < 2, using Eqs. 2.5, 4.5 and 4.7, give Eq. 4.15. □
Corollary 4.8
There exists an equivalent functionalL1 ∼ Esuch that,
and
for some positive constants λand c.
Proof
Using Eqs. 2.5 and 4.10 we conclude that, for any t ≥ t1,
By letting L1(t) = L(t) + cE(t) and combining Eqs. 4.14 and 4.21, Eq. 4.19 is established. Similar calculations hold, for m < 2, to obtain (4.20). □
5 Stability
In this section we state and prove our main result. We start with the following lemmas.
Lemma 5.1
Assume that (A1) and (A2) hold andm ≥ 2. Then, the energy functional satisfies the following estimate
Proof
Let F(t) = L(t) + ψ3(t), then using Eq. 4.9, we obtain
Using Eqs. 2.5 and 5.2, we obtain
where b is a positive constant. Therefore,
where F1(t) = F(t) + cE(t) ∼ E. □
Lemma 5.2
Assume that (A1) and (A2) hold and 1 < m < 2. Then, the energy functional satisfies the following estimate
Proof
Let F(t) = L(t) + ψ3(t), then using (4.9) and (4.15), we obtain
By multiplying Eq. 5.5 by Eq(t), q > 0,and using Young’s inequality, we get
By choosing \(q=\frac {2-m}{2m-2}\) and taking ε small, Eq. 5.6 yields
Let F2(t) = Eq(t)F(t) + CE(t) then Eqs. 2.5, 4.13 and 5.7, lead to
Therefore,
which gives Eq. 5.4 since \(1+q=\frac {m}{2m-2}\). □
Remark 5.3
Using Hölder’s inequality and Eq. 5.4, we obtain, for 1 < m < 2,
Let’s define
Lemma 5.4
Under the assumptions (A1) and (A2) , we have the following estimates
where p ∈ (0, 1) and\(\overline {G}\)is an extension ofGsuch that\(\overline {G}\)is strictly increasing and strictly convex C2function on (0, ∞); see Remark 2.1.
Proof
First, we define the following quantity
Using Eqs. 2.4 and 5.1, we obtain
Also, we can choose p so small that, for all t > t1,
Since G is strictly convex on (0, r] and G(0) = 0, then
The use of Eqs. 2.2, 5.15, 5.16 and Jensen’s inequality yields
This gives Eq. 5.12 when m ≥ 2. In the case 1 < m < 2 and for the proof of Eq. 5.13, we define the following
then using Eqs. 2.4 and 5.10, we easily see that
then choosing p ∈ (0, 1) small enough so that Eq. 5.15 holds and
The use of Eqs. 2.2, 5.16, 5.18 and Jensen’s inequality leads to
This implies that
□
Theorem 5.5
Let\((u_{0},u_{1})\in {H_{0}^{1}}({\Omega })\times L^{2}({\Omega })\)begiven. Assume that (A1) and (A2) hold andm ≥ 2. Then there exist strictly positive constantsc1,c2,k1andk2such that the solution of Eq. 1.1satisfies, for allt ≥ t1,
where\(G_{1}(t)={\int }_{t}^{r_{1}}\frac {1}{sG^{\prime }(s)}ds.\)
Proof
-
Case 1: G is linear.
Using Eqs. 2.2 and 2.5, we get
Multiplying (4.19) by ξ(t) and using Eq. 5.22, we obtain
which gives, as ξ(t) is non-increasing,
Hence, using the fact that ξL + 2cE ∼ E, we easily obtain
Case 2: G is non-linear.
Using (4.19) and (5.12), we obtain
Then, the functional \(\mathcal {F}_{1}, \) defined by
satisfies, for some α1, α2 > 0.
and
Let \(\overline {G}^{*}\) be the convex conjugate of \(\overline {G}\) in the sense of Young [2], then
and \(\overline {G}^{*}\) satisfies the following generalized Young inequality
So, with \(A=\overline {G}^{\prime }\left (\varepsilon _{0}\frac {E^{\prime }(t)}{E(0)}\right )\) and \(B=\overline {G}^{-1}\left (\frac {pI(t)}{\xi (t)}\right )\) and using Eqs. 2.5 and 5.27–5.29, we arrive at
So, multiplying (5.30) by ξ(t) and using (5.11) and the fact that \(\varepsilon _{0}\frac {E(t)}{E(0)}<r\), \(\overline {G}^{\prime }\left (\varepsilon _{0}\frac {E(t)}{E(0)}\right )=G^{\prime }\left (\varepsilon _{0}\frac {E(t)}{E(0)}\right )\), we get
Consequently, with a suitable choice of ε0, we obtain, for all t ≥ t1,
where \(\mathcal {F}_{2}=\xi \mathcal {F}_{1}+c E \sim E\) and G2(t) = tG′(ε0t). Since \(G^{\prime }_{2}(t)=G^{\prime }(\varepsilon _{0}t)+\varepsilon _{0}t G^{\prime \prime }(\varepsilon _{0}t), \)then, using the strict convexity of G on (0, r],we find that \(G_{2}^{\prime }(t), G_{2}(t)>0\) on (0,1]. Thus, taking in account (5.26) and (5.31), we easily see that
satisfies
and, for some k1 > 0,we have
Then, the integration over (t1, t) yields
Hence, by an approprite change of variable, we get
Thus, we have
where \(G_{1}(t)={\int }_{t}^{r_{1}}\frac {1}{sG^{\prime }(s)}ds.\) Here, we have used the fact that G1 is strictly decreasing on (0, r]. Therefore (5.21) is established by virtue of Eq. 5.32. □
Remark 5.6
The decay rate of E(t) given by Eq. 5.21 is optimal because it is consistent with the decay rate of g(t) given by Eq. 2.2. In fact,
where \(G_{0}(t)={{\int }_{t}^{r}}\frac {1}{G(s)}\).
Using the properties of G, G0 and G1, we can see that
Using the fact that G1 is decreasing, we have
By putting τ = G0(t), we obtain
Therefore,
This shows that Eq. 5.21 provides the best decay rates expected under the very general assumption (2.2).
Theorem 5.7
Let\((u_{0},u_{1})\in {H_{0}^{1}}({\Omega })\times L^{2}({\Omega })\)begiven. Assume that (A1) and (A2) hold and 1 < m < 2. Then, there exist postive constantsc, m5, m6such that
where\(W_{2}(\tau )= \tau ^{\frac {m}{2m-2}} G^{\prime }(\varepsilon _{1} \tau ).\)
Proof
-
Case 1: G is linear
Multiplying (4.20) by ξ(t) and using Eq. 5.22, we obtain
which gives, as ξ(t) is non-increasing,
where \(\mathcal {L}(t)=\xi (t)L(t)+cE(t)\sim E\). By multiplying the last inequality by Eq(t), q > 0, recalling (2.5), and using Young’s inequality, we get
By choosing \(q=\frac {2-m}{2m-2}\), Eq. 5.37 yields
Let \(\mathcal {L}_{1}(t)=E^{q}(t) \mathcal {L}(t)+c(\varepsilon )E, \)then using Eqs. 2.5, 5.38, the fact that \(\mathcal {L}_{1}\sim E\), and choosing ε small enough, we get
The last inequality together with the equivalence relation (\(\mathcal {L}_{1} \sim E\)) give (5.34).
Case 2:Gis nonlinear
Using Eqs. 4.19 and 5.13, we obtain
we find that the functional L1,defined by
satisfies, for some β1, β2 > 0.
and
So, with \(A=G^{\prime }\left (\frac {\varepsilon _{1}}{(t-t_{1})^{\frac {2m-2}{m}}} \cdot \frac {E(t)}{E(0)}\right )\) and \(B=\left (\overline {G}\right )^{-1}\left (\frac {pI(t)}{(t-t_{1})^{\frac {2m-2}{m}}\xi (t)}\right )\) and using Eqs. 2.5, 5.28, 5.29 and 5.42 yields
By multiplying the last inequality by \(\xi (t) E^{\frac {2-m}{2m-2}}(t), \) using Eqs. 2.5, 5.4, 5.11 and Young’s inequality, we get
The last inequality becomes
Let \(L_{2}(t)=\xi (t) E^{\frac {2-m}{2m-2}}(t) L_{1}(t)+\left (c(\varepsilon )+c\right )E, \) then using Eqs. 2.5, 4.13, 5.45 and choosing ε1 and ε small enough, we get
for some m1 > 0. Then, we have, for m2 = m1E(0),
An integration of Eq. 5.47 yields
Using the fact that G′, G″ > 0 and the non-increasingness of E, we deduce that the map \(t \mapsto \frac {E^{\frac {m}{2m-2}}(t)}{E(0)}G^{\prime }\left (\frac {\varepsilon _{1}}{(t-t_{1})^{\frac {2m-2}{m}}} \cdot \frac {E(t)}{E(0)}\right )\) is non-increasing and consequently, we have
Multiplying each side of Eq. 5.49 by \(\frac {1}{(t-t_{1})}\), we have
Next, we set \(W_{2}(\tau )=\tau ^{\frac {m}{2m-2}} G^{\prime }(\tau )\) which is strictly increasing, then we obtain,
Finally, for two positive constants m5 and m6, we obtain
This finishes the proof. □
Example 5.8
The following examples illustrate our results:
- 1.
G is linear and m ≥ 2
Let g(t) = ae−b(1 + t), where b > 0 and a > 0 is small enough so that Eq. 2.1 is satisfied, then g′(t) = −ξ(t)G(g(t)) where G(t) = t and ξ(t) = b. Therefore, we can use Eq. 5.20 to deduce
$$ E(t) \leq c_{1} e^{-c_{2}t} $$(5.53)which is the exponential decay.
- 2.
G is non-linear and m ≥ 2
Let \(g(t)=a e^{-t^{p}}, \)where 0 < p < 1 and a > 0 is small enough so that g satisfies (2.1), then g′(t) = −ξ(t)G (g(t)) where ξ(t) = 1 and \(G(t)=\frac {p^{t}}{\left (ln (a/t)\right )^{1/p-1}}\). Since
$$\begin{array}{lllll} &G^{\prime}(t)=\frac{(1-p)+p ln (a/t)}{\left( ln (a/t)\right)^{1/p}}\\ &\hspace{0.45in}\text{and}\\ &G^{\prime\prime}(t)=\frac{(1-p) \left( ln (a/t)+ 1/p \right)}{\left( ln (a/t)\right)^{\frac{1}{p + 1}}}. \end{array} $$then the function G satisfies the condition (A1) on (0, r] for any 0 < r < a.
$$\begin{array}{lllll} &G_{1}(t)={{\int}_{t}^{r}}\frac{1}{sG^{\prime}(s)}ds={{\int}_{t}^{r}}\frac{\left[\ln{\frac{a}{s}}\right]^{\frac{1}{p}}}{s\left[1-p+p\ln{\frac{a}{s}}\right]}ds\\ &\hspace{0.3in}={\int}_{\ln{\frac{a}{r}}}^{\ln{\frac{a}{t}}}\frac{u^{\frac{1}{p}}}{1-p+pu}du\\ &\hspace{0.3in}=\frac{1}{p}{\int}_{\ln{\frac{a}{r}}}^{\ln{\frac{a}{t}}} u^{\frac{1}{p}-1}\left[\frac{u}{\frac{1-p}{p}+u}\right]du\\ &\hspace{0.3in}\le \frac{1}{p}{\int}_{\ln{\frac{a}{r}}}^{\ln{\frac{a}{t}}} u^{\frac{1}{p}-1}du \le \left( \ln{\frac{a}{t}}\right)^{\frac{1}{p}}. \end{array} $$Then, Eq. 5.21 gives
$$ E(t)\leq k e^{-k t^{p}}. $$(5.54) - 3.
G is linear and 1 < m < 2
Let g(t) = ae−b(1 + t),where b > 0 and a > 0 is small enough so that Eq. 2.1 is satisfied, then g′(t) = −ξ(t)G(g(t)) where G(t) = t and ξ(t) = b. Therefore, applying (5.34), we obtain
$$ E(t)\leq \left[ \frac{1}{1+t} \right]^{\frac{2m-2}{2-m}}. $$(5.55) - 4.
G is non-linear and 1 < m < 2
Let \(g(t)=\frac {a}{(1+t)^{2}}\), a is chosen so that hypothesis (2.1) remains valid. Then
$$g^{\prime}(t)=-b G(g(t)),\hspace{0.2in}\text{with}\hspace{0.2in}G(s)=s^{\frac{3}{2}}, $$where b is a fixed constant. Then, \(W_{2}(t)=c t^{\frac {2m-2}{2m-1}}.\) Therefore, applying (5.35), we get
$$ E(t)\le \frac{1}{\left( t-t_{1}\right)^{\frac{-3m^{2}+ 6m-2}{m(2m-1)}}}, $$(5.56)for \(1<m<1+\frac {\sqrt {3}}{3}\), \(\frac {-3m^{2}+ 6m-2}{m(2m-1)}>0.\)
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Acknowledgments
The authors thank an anonymous referee for his/her very careful reading and valuable suggestions. This work was funded by KFUPM under Project #IN161006.
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This work was funded by KFUPM under Project #IN161006.
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Belhannache, F., Algharabli, M.M. & Messaoudi, S.A. Asymptotic Stability for a Viscoelastic Equation with Nonlinear Damping and Very General Type of Relaxation Functions. J Dyn Control Syst 26, 45–67 (2020). https://doi.org/10.1007/s10883-019-9429-z
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DOI: https://doi.org/10.1007/s10883-019-9429-z