Abstract
In this paper we consider a nonlinear viscoelastic beam with a linear delay term and infinite memory term. The well posedness of solutions is proved using the semigroup method. We establish a general decay results by using minimal and general conditions on the relaxation function, from which the usual exponential and polynomial decay rates are only special cases.
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1 Introduction
In this paper, we consider the following one-dimensional version of a system which describes the vibrations of shallow shells with time delay and infinite memory term:
where \(\mathscr {Q}=I \times \mathbb {R}_{+}\), \(I=]0,L[\) is an interval, \(\partial I \) its boundary (\(\partial I =\{0\}\cup \{L\}\)), \(\Sigma =\left\{ 0\right\} \times \mathbb {R}_{+}\cup \left\{ L\right\} \times \mathbb {R}_{+}\), \(\mathscr {D}\) is the operator \(\mathscr {I}-{\partial }_{x}^{2}\). The functions f and g are defined by:
In (1), subscripts mean partial derivatives, the space variable x runs in the interval \(0< x < L\) and t denotes the positive time variable. The functions \(u = u(x, t)\) and \(w = w(x, t)\) are, respectively, the longitudinal and transversal displacements of the beam at the point x at time t. Additionally, \(\mu \) is a constant, \(0< \mu < 1\) and \(k= k(x)\) represents the curvature of the beam at the point x.
In the system (1), \(\alpha _{1}u _{t}\) represents a frictional damping. The time delay is given by \(\alpha _{2}u_{t}(t-\tau )\), where \(\alpha _{1}\), \(\alpha _{2}\), \(\tau \) are positive constants.
In (1), \((h*u)(t)\) is defined by
The viscoelastic damping term that appears in the equations describes the relationship between the stress and the history of the strain in the beam, according to Boltzmann theory. The function h represents the kernel of the memory term or the relaxation function.
The system (1) is subjected with the boundary conditions
and the initial conditions
The main purpose about problem (1)–(5) is to deal with the well posedness and asymptotic behavior of solutions. Before stating and proving our results, let us recall some other results related to our work.
Several authors have studied the Mindlin–Timoshenko system of equations (see, e.g., [16]). This Model is a widely used and fairly complete mathematical model for describing the transverse vibrations of beams. It is a more accurate model than the Euler-Bernoulli one, since it also takes into account transverse shear effects.
For a beam of length \(L > 0\), this one-dimensional nonlinear system reads as
where \(Q = (0,L) \times (0, T )\) and T is a given positive time. Here, the unknown \(\phi = \phi (x,t)\) represent the angle of rotation. The parameter k is the so called modulus of elasticity in shear. It is given by the expression \(k = \widehat{k}Eh_{0}/2 (1 + \epsilon )\), where \(\widehat{k}\) is a shear correction coefficient, E is the Young’s modulus and \(\epsilon \) is the Poisson’s ratio, \(0< \epsilon < 1/2\).
For Mindlin–Timoshenko system, there is a large literature, addressing problems of existence, uniqueness and asymptotic behavior in time when some damping effects are considered, as well as some other important properties (see [13, 26, 28] and references therein).
When one assumes the linear filament of the beam to remain orthogonal to the deformed middle surface, the transverse shear effects are neglected, and one obtains, from the Mindlin–Timoshenko system of equations, the following von Kármán system (see [28]).
There is also an extensive literature about system (7) (see [13, 18, 26, 27, 38, 51, 53,54,55,56] and references therein).
Lagnese and Leugering [27] considered a one-dimensional version of the von Kármán system describing the planar motion of a uniform prismatic beam of length L. More precisely, in [27] the following system was considered:
In [27], suitable dissipative boundary conditions at \(x = 0\), \(x = L\) and initial conditions at \(t = 0\) were given and the stabilization problem was studied.
In [4], Araruna et al. have showed how the so called von Kármán model (8) can be obtained as a singular limit of a modified Mindlin–Timoshenko system (6) when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin–Timoshenko system decays exponentially, uniformly with respect to the parameter k. As \(k\longrightarrow \infty \), the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.
The subject of stability of von Kármán system has received a lot of attention in the last years, see [11, 12, 18, 18, 25, 26, 33, 39, 49, 50, 52] and references therein.
Delay effects are very important because most natural phenomena are in many cases very complicated and do not depend only on the current state but also on the past history of the system. The presence of delay can be a source of instability. In recent years, the stabilization of PDEs with delay effects has draw attention for many author and become an active area of research, see [11, 15, 24, 46,47,48, 57, 59,60,61].
For the stability of other kind of wave equation, let us mention the following problem:
where \(\Omega \) is a bounded domain in \(\mathbb {R}^{n}\), \(n \in \mathbb {N}\), with a smooth boundary \(\partial \Omega =\Gamma \), h is a positive non-increasing function defined on \(\mathbb {R}^{n}\), \(h_{1}\) and \(h_{2}\) are two functions, \(\tau > 0\) is a time delay, \(\alpha _{1}\) and \(\alpha _{2}\) are positive real numbers and the initial data \(( u_{0}, u_{1}, f_{0})\) belong to a suitable function space.
In the case \(h \equiv 0\), problem (9) has been studied by many authors (see [6,7,8, 10, 46, 61]).
For a wider class of relaxation functions, Messaoudi [36, 37] considered
for \(\gamma > 0\) and \(b = 0\) or \(b = 1\), and the relaxation function satisfies
where \(\zeta \) is a differentiable nonincreasing positive function. He established a more general decay result, from which the usual exponential and polynomial decay results are only special cases. Such a condition was then employed in a series of papers, see for instance [3, 22, 23, 41, 42, 52].
Recently, Mustafa and Messaoudi [45] studied the problem (10) with \(b = 0\) for the relaxation functions satisfying
where H is a nonnegative function, with \(H(0) = H^{\prime }(0)= 0\) and H is strictly increasing and strictly convex on ]0, k[ for some \(k_{0}> 0\). The authors showed a general relation between the decay rate for the energy and that of the relaxation function h without imposing restrictive assumptions on the behavior of h at infinity. On the other hand, a condition of the form (12) where H is a convex function satisfying some smoothness properties, was introduced by Alabau-Boussouira and Cannarsa [2] and used then by several authors with different approaches. We refer to [32] where not only general but also optimal result was established by Lasiecka and Wang.
The main objective of this work is to investigate the problem (1) with the following very general class of relaxation functions
where H is increasing and convex without any additional constraints on H and the coefficients. We will establish a general decay rate for the energy associated to the system for linear damping, time delay terms and finite memory. We would like to see the influence of frictional and viscoelastic dampings on the rate of decay of solutions in the presence of linear degenerate delay term.
To prove decay estimates, we shall pursue a strategy based on an adaptation of non linear differential inequalities technique developed in [40, 43, 44] and we use a perturbed energy method and some properties of convex functions which were introduced and developed by many authors [1, 9, 14, 17, 30, 31, 34].
Our work is organized as follows. In the next section, we prepare some material needed in the proof of our result, like some lemmas (Poincaré’s and Young’s inequalities) and some useful notations. We introduce the different functionals by which we modify the classical energy to get an equivalent useful one. In Sect. 4, we state and prove the well-posedness of the problem. Finally, in Sect. 5, we will prove our main results concerning the exponential decay of the energy associated to the solutions of the problem.
2 Statement of results
In order to deal with the delay feedback term, motivated by [46, 47], we define the following new dependent variables \(\eta \) and z:
consequently, we obtain
clearly, (14) gives
where \(z_{p}=\partial _{p}z\) and \(\eta _{s}=\partial _{s}\eta \).
Therefore, problem (1)–(5) is equivalent to
where \(l=1-\int _{0}^{\infty }h(s)ds\), with boundary conditions
and initial conditions
In what follow, we assume that the function k belong to the sobolev space \(H^{1}(0,L)\).
3 Preliminaries
In this section, we state our stability results for problem (17)–(19). For this purpose, we start with the following hypotheses:
(\(\mathscr {H}_{1}\)) \(h: \mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) is a non-increasing differentiable function such that \(h (0)>0\) and
Also assume that there exist a positive constant \(\alpha \)such that
(\(\mathscr {H}_{1}\)) There exists an increasing strictly convex function \(G:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) of class \(C^1 (\mathbb {R}_{+})\cap C^2(]0,+\infty [)\) satisfying
such that
Remark 1
The condition (22) introduced in [19] is satisfied by any positive function h of class \(C^{1}(\mathbb {R}_{+})\) with \(h^{\prime }< 0\) and h is integrable on \(\mathbb {R}_{+}\) (see [19,20,21] for explicit examples).
C and c denote some general positive constants, which may be different in different estimates.
3.1 Functional setting and assumptions.
In order to prove the well-posedness of (17) by using the semigroups theory, we introduce some functional spaces.
Let us introduce the energy space \(\mathscr {H}\) by
where \(L_{h}(I)\) is the weighted Sobolev space defined by
The space \(L_{h}(I)\) is endowed with the inner product
Also, we denote by \(\left\langle \left\langle .,.\right\rangle \right\rangle \) the natural inner product on the space \(L^{2}(I\times \left( 0,1\right) )\), we note that the norms
are equivalent in \(L^{2}(I\times \left( 0,1\right) ).\)
Let \(\xi \) be any positive number which satisfy
Also, we define
\(\mathscr {H}\) is endowed with the norm
where
and
3.2 Energy identity
We start by the following lemma:
Lemma 1
For \(\left( h,\phi \right) \in \left( \mathscr {C}^{1}\cap L^{1}\right) \left( \mathbb {R}_{+}\right) \times \mathscr {C}^{1}\left( \mathbb {R}\right) \), we have
Lemma 2
Assume that \((\psi ,\psi _{t},\eta ,\eta _{t},z)\) is a strong solution of the problem (17)-(19). Then we have
Proof
We multiply the third equation in (17) by \(\frac{\xi }{\tau }z\) and integrate the result over \((0,L) \times (0,1)\) with respect to p and x, respectively, to get
which gives (25). \(\square \)
We define the energy associated with the solution of system (17)–(19) by
where \(\xi \) is a positive constant such that
and \(\alpha _{1}\) and \(\alpha _{1}\) satisfying
Lemma 3
Assume that \((\psi ,\psi _{t},\eta ,\eta _{t},z)\) is a strong solution of the problem (17)–(19). Then the derivative of \(\mathscr {E}(t)\) satisfies
Moreover, for all \(t \ge 0\), we have
Proof
Multiplying the first equation in (17) by \(u_{t},\) the second by \(w_{t}\) and the third by \(\xi z(p),\) integrating by part and using boundary condition in (18) and Lemma 2 yields (30). \(\square \)
Lemma 4
(Jensen inequality) Let F be a convex function on [a, b], \(r_{1}:\Omega \rightarrow [a,b]\) and \(r_{2}\) are integrable functions on \(\Omega ,\) \(r_{2}(x)\ge 0\), and \(\int _{\Omega }r_{2}(x)dx=k_{0}>0\), then Jensen’s inequality states that
4 Global well-posedness
In this section we show the existence and regularity of solutions of the one dimensional viscoelastic Marguerre–Vlasov system (17)–(19).
Then problem (17)–(19) is reduced to the following problem for an abstract first-order evolutionary equation:
where \(\mathscr {U} = (u,u_{t},w,w_{t}, z,\eta )^{T}\), and
with the domain
Lemma 5
The operator \(\mathscr {A}\) defined in (32) is the infinitesimal generator of a \(C_{0}\)-semigroup in \(\mathscr {H}\).
Proof
For all \(\mathscr {U}(t) \in D(\mathscr {A})\), one has
Integrating by parts, we have
and
Plugging (34) and (35) in (33), we get
which implies that \(\mathscr {A}\) is dissipative.
Next we will prove that the operator \(\left( \mathscr {I}-\mathscr {A}\right) \): \(D(\mathscr {A})\longrightarrow \mathscr {H}\) is onto, that is, given \(\mathscr {F}=(f_{1},f_{2},f_{3},f_{4},f_{5},f_{6})\in \mathscr {H}\), we seek \(\mathscr {U} = (u,u_{t},w,w_{t}, z,\eta )^{T} \in D(\mathscr {A})\) such that
Equivalently, one must consider the system given by
By integrating the Eqs. (42) and (43), we obtain
and
Plugging (44) and (45) in (39) and (41) and keeping in mind that \(u_{t}=u-f_{1}\) and \(w_{t}=w-f_{3}\) we find that u and w fulfil the equations
where
Since \(x\mapsto \left( f_{1},f_{5}\right) \in H_{0}^{2}(I)\times H_{0}^{2}(I) \) then \(\left( f_{1xxxx},f_{5xxxx}\right) \in H^{-2}(I)\times H^{-2}(I)\). To show that \(\beta _{1}\in H^{-2}(I)\), we have to show that
Applying Cauchy–Shwarz inequality and Fubini’s theorem, we get
Thus
Therefore, by Lax–Milgram Theorem, system (38) admits a unique solution \(\mathscr {U}\in D(\mathscr {A})\). This means that (37) holds and consequently \(I-\mathscr {A}\) is onto. Thus, by using the Lumer–Phillips [35, Theorem 4.3], we deduce that the operator \(\mathscr {A}\) generates a \(C_{0}\) semigroup of contractions in \(\mathscr {H}\). \(\square \)
Lemma 6
The operator \(\mathscr {B}\) defined in (32) is locally Lipschitz in \(\mathscr {H}\).
Proof
Let \(\widetilde{\mathscr {U}}=(u,u_{t},w,w_{t},z,\eta )\) and \(\widetilde{\mathscr {U}}=(\widetilde{u},\widetilde{u}_{t},\widetilde{w},\widetilde{w}_{t},\widetilde{z},\widetilde{\eta })\) two elements of \(\mathscr {H}{\normalsize .}\) A direct calculation shows that
where
and
So we have to estimate F and G in \(L^{2}(I)\) and \(H_{0}^{1}(I)\) norm respectively.
Since \(k\in H^{1}(I)\), we can use the embedding \(H^{1}(I)\hookrightarrow L^{\infty }(I)\) to prove
Now, let
Taking into account that the operator \(\mathscr {D}^{-1}\partial _{x}\) is bounded from \(L^{2}(I)\) into \(H_{0}^{1}\left( I\right) \), we can write
Adding and subtracting the term \(\widetilde{w}_{x}\left( u_{x}+\frac{1}{2}w^{2}+k(x)w\right) \) inside the norm on the right hand side of (27) and proceed with the same manner, we find that
Finally, let
Similarly, have
Then the operator \(\mathscr {B}\) is locally Lipschitz in \(\mathscr {H}\). So problem (17)–(19) admits a local solution. The proof is hence complete. \(\square \)
The boundedness of the energy in (30) allows to extend the solution on [0, T] for an any arbitrary \(T>0,\) so we have shown:
Theorem 1
Assume that \(\left( \textbf{H}_{1}\right) \) holds. Let \(\mathscr {U}_{0}\in D(\mathscr {A}),\) then (17)–(19) has a unique solution
5 General decay
In this section we consider a wider class of kernel functions, and we establish a general decay result, where exponential and polynomial decay rates are special cases.
The main result of general decay is the following.
Theorem 2
Assume that (20) hold such that (21) hold or there exists a positive constant M such that
then there exists positive constants \(c^{\prime },c^{\prime \prime }\) and \(\epsilon _{0}\) for which \(\mathscr {E}\) satisfies
or
where
Remark 2
The previous theorem shows that exponential decay holds when (20) holds, otherwise, if (21) holds, we get a weak decay of energy. For precise examples illustrating (57) see [19,20,21].
To prove Theorem 2, we need some useful lemmas.
Lemma 7
The following inequalities holds
Proof
For inequality (58), we have
Cauchy–Schwarz inequality leads to
Similarly, we prove (59) by replacing \(\sqrt{h(t-s)}\) by \(\sqrt{-h^{\prime }(t-s)}.\) \(\square \)
Let \(\mathscr {F}\) be the functional defined by
where
\(\mathscr {E}(t)\) is defined in (26), \(\lambda > 0\), \(\delta _{1}\), \(\delta _{2}\) and \(\delta _{3}\) are positive constants that will be chosen later.
The following proposition gives the equivalence between \(\mathscr {E}(t)\) and the functional \(\mathscr {F}(t)\).
Proposition 1
Assume that \(\left( \textbf{H}_{1}\right) \) holds, then there exists two positive constants \(\beta _{1}\), \(\beta _{2}\) such that
Proof
To compare \(\mathscr {F}(t)\) with \(\mathscr {E}(t)\), we have to estimate the terms \(\mathscr {I}\left( t\right) \), \(\mathscr {J}\left( t\right) \) and \(\mathscr {K}\left( t\right) \) of the right hand side of (60) and show that.
From (61), (62) and (63), we obtain
\(\bullet \) Estimate for \(\mathscr {I}_{1}(t)\)
Using Poincaré’s and (72, we obtain
where \( c_{1}\) is a positive constant.
\(\bullet \) Estimate for \(\mathscr {I}_{2}(t)\)
Using Poincaré’s and (58, we obtain
where \( c_{2}\) is a positive constant.
\(\bullet \) Estimate for \(\mathscr {I}_{3}(t):=\int _{0}^{L}\int _{0}^{1}e^{-2\tau p}z^{2}dpdx\)
Since \(\mathscr {I}_{3}(t)\) defines a norm in \(L^{2}(0,L;L^{2}(0,1))\) which is equivalent to the one induced by \(L^{2}(0,L;L^{2}(0,1))\), then we have
where \( c_{3}\) is a positive constant.
According to (65), (66) and (67), we have
where
Therefore, we obtain
that is
So, we choose \(\lambda \) large enough such that \(\beta _{1} = \lambda - c^{*}> 0\), \(\beta _{2} = \lambda + c^{*}> 0\). Then (64) holds true.
This completes the proof. \(\square \)
In order to proof the main theorem, we need some additionals lemmas.
Lemma 8
Suppose that \((\psi ,\psi _{t},\eta ,\eta _{t},z)\) is the solution of (17)–(19). Then the derivative of the functional \(\mathscr {I}_{1}(t)\) satisfies
where \(\varepsilon \) is an arbitrary positive constant.
Proof
Multiplying the first equation in (17) by u, the second by w and taking account that \(\frac{d}{dt}\int _{0}^{L}h_{t}hdx=\int _{0}^{L}\left( h_{tt}h+h_{t}^{2}\right) dx,\) then integrating by part, we get
Using Young’s and Poincaré’s inequalities, the two last terms are estimated by
and using (58), we arrive at
Now to estimate \(\int _{0}^{L}u^{2}dx\), we have
Plugging (70), (71) and (72) in (69) this proves (68). \(\square \)
Lemma 9
Assume that \((\psi ,\psi _{t},\eta ,\eta _{t},z)\) is the solution of (17)–(19). Then the derivative of the functional \(\mathscr {I}_{2}(t)\) satisfies
where \(\varepsilon \) is an arbitrary positive constant.
Proof
Multiplying the second equation in (18) by \(\int _{0}^{\infty }h(s)\mu (s)ds\) and integrating by parts, we get
For the first term in the right hand side of (74), we have
In the other hand, we also have
Plugging (76) in (75), we infer
Integrating by parts and applying Young’s inequality and (59), then Poincaré’s inequality, we get
Using the embedding \(H^{1}(I)\hookrightarrow L^{\infty }\left( I\right) \), we estimate \(-\int _{0}^{L}\left[ f(u,w)\right] _{x}\int _{0}^{\infty }h(s)\mu (s)dsdx\) as follows
Similarly, we estimate \(\int _{0}^{L}kg(u,w)wdx\) using the umbedding \(H^{1}(I)\hookrightarrow L^{\infty }\left( I\right) \).
Gathering (77), (78) and (79), (73) is proven. \(\square \)
Lemma 10
Suppose that \((\psi ,\psi _{t},\eta ,\eta _{t},z)\) is the solution of (17)–(19). Then the time derivative of the functional \(\mathscr {I}_{3}(t)\) satisfies
Proof
Multiplying the third equation in (17) by \(e^{-2\tau p}z\) and integrating over \(I\times \left( 0,1\right) \), we arrive at
which gives (80) \(\square \)
Proposition 2
Assume that \(\left( \textbf{H}_{1}\right) \) and \(\left( \textbf{H}_{2}\right) \) hold, then there exists two positive constants \(\beta _{1},\beta _{2}\) such that
Proof
By using (60) and combining (30), (68), (73) and (80), we get
We want to impose suitable conditions on the different parameters so that the coefficients on the right hand side of (82) are all strictly negative. That is to obtain the following inequalities
We observe that (83) and (84) will be satisfied if we choose \(\varepsilon > 0\) small enough and such that
To make (84) and (85) hold we can choose
Concerning (87), (88) and (89), we pick
This completes the proof. \(\square \)
We consider the following two cases.
Case I. H(t) is linear:
By multiplying (81) by \(\xi (t)\) and using (30), we get
which gives, as \(\xi \) is nonincreasing,
Hence, using the fact that \(\mathscr {F}(t)\xi (t)+c\mathscr {E}(t)\) is equivalent to \(\mathscr {E}(t)\), it is easy to see that
for some \(\beta _{1} > 0\). Then
from which we deduce
for some \(\gamma _{2}>0.\)
Furthermore, using the continuity and boundedness of \(\mathscr {E} (t)\) in \([0, t_{1}]\), we get
Case II. H(t) is nonlinear:
Next, with \(f(t) =\int _{t}^{\infty }h(s)ds\), we use the functional
Lemma 11
Suppose that \((\psi ,\psi _{t},\eta ,\eta _{t},z)\) is the solution of ((17))–(19). The functional \(\mathscr {K}\) defined by (90) satisfies, for any \(\varepsilon >0\), the estimate
Proof
By Young’s inequality and the fact \(f^{\prime }(t)=-h(t)\), we see that
But
Combining (92) and (93), we obtain (91). \(\square \)
Let us introduce the functional
where \(\sigma \) is a positive constant. Then we have
Therefore, it is always possible to pick \(N_{1}\) (in 82) and h large enough to get
Integrating over \((t_{0},\infty )\), we get
Next, let us define the functional \(\mathscr {L}(t)\)
where \(q>0.\) Thanks to (94), we can always choose q such that
Next we define
Observe that
for some positive constant C.
Since H is strictly convexe on (0, r] and \(H(0)=0\) we have
Using \(\left( \textbf{H}_{2}\right) \), we get:
Keeping in mind (95) and applying inequality (96) for \(\theta :=\mathscr {L}(t)\) and \(x=h(s)\), yields
Applying Jensen’s inequality in (31) for \(r_{1}(t)=\mathscr {L}(t)h(s)\) and \(r_{2}(s)= q\left\| {\psi }_{xx}(t)-{\psi }_{xx}\right. \left. (t-s)\right\| ^{2},\) we obtain
where \(\overline{H}\) is an extention of H such that \(\overline{H}\) is strictly increasing and strictly convexe \(\mathscr {C}^{2}\) function on \((0,\infty )\) and this leads to
So (81) becomes
Let \(\epsilon _{0}<r\), using the fact that \(\mathscr {E}^{\prime }\le 0,\) \(\overline{H}^{\prime }>0,\overline{H}^{\prime \prime }>0\), we observe that the functional \(\mathscr {N}\) defined by
is equivalent to \(\mathscr {E}\).
Using (98), we find that \(\mathscr {N}\) satisfies
Let us denote by \(G^{*}\) the conjugate function of the convex function G defined by \(G^{*}(s)=Sup_{t\in \mathbb {R}^{+}}(st-G(t))\), then
and, thanks to the arguments given in [5, 9, 14, 29, 30]
This and the definition of H give
Taking \(s:=\frac{C_{2}}{q}\overline{H}^{\prime }\left( \epsilon _{0} \frac{\mathscr {E}(t)}{\mathscr {E}(0)}\right) \) and \(t:=\overline{H} ^{-1}\left( \frac{q\mathscr {L}_{g}(t)}{\xi (t)}\right) \) in (100), then making use of (99), (100) and (101), we arrive at
Next, multiplying (102) by \(\xi (t)\) and using the fact that \(\epsilon _{0}\frac{\mathscr {E}(t)}{\mathscr {E}(0)}<r,\) \(\overline{H} ^{\prime }\left( \epsilon _{0}\frac{\mathscr {E}(t)}{\mathscr {E}(0)}\right) =H^{\prime }\left( \epsilon _{0}\frac{\mathscr {E}(t)}{\mathscr {E}(0)}\right) \), we get
where c is a positive constant [58].
Now, let us define the functional \(\widetilde{\mathscr {N}}\)
It is not difficult to see that there exist positive constants \(\rho _{1}\) and \(\rho _{2}\) for which we have
Consequently, with an appropriate choice of \(\epsilon _{0}\), then there exists a positive constant k such that
where \(H_{2}(s)=sH^{\prime }(\epsilon _{0}s).\)
Since \(H_{2}^{\prime }(s)=H^{\prime }(\epsilon _{0}s)+\epsilon _{0}sH^{\prime \prime }(\epsilon _{0}s)\), we use the strict convexity of H on [0, r), we observe that \(H_{2}>0,\) \(H_{2}^{\prime }>0\) on (0, r].
Defining now
thanks to (103) and (104) we have \( \mathscr {E}\sim \mathscr {R}\) and for a positive constant \(\widetilde{k}\)
Then, integrating over \((t_{0},t)\) yields
and this leads to
which gives us
where \(H_{1}(t)=\int _{t}^{r}\frac{ds}{sH^{\prime }(s)}\).
This completes the proof.
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The author would like to thank the anonymous referees for their valuable suggested comments and would like to express his gratitude to DGRSDT for the financial support. The author is grateful to the editors for their help.
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Khemmoudj, A. General decay of the solution to a nonlinear viscoelastic beam with delay. Partial Differ. Equ. Appl. 4, 20 (2023). https://doi.org/10.1007/s42985-023-00238-y
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DOI: https://doi.org/10.1007/s42985-023-00238-y