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General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities
Boundary Value Problems volume 2022, Article number: 23 (2022)
Abstract
In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.
1 Introduction
In this work, we study the following viscoelastic von Karman equation with damping, delay, and source terms of variable exponents:
where Ω is a bounded domain in \({\mathbb{R}}^{2}\) complementing a smooth boundary ∂Ω, ν is the unit normal vector outward to ∂Ω, \(\alpha >0 \), \(\beta \in {\mathbb{R}} \), \(\gamma >0 \), \(\tau >0 \) is time delay, \(u_{0}\), \(u_{1}\), and \(j_{0}\) are given initial data, and the von Karman bracket \([\cdot ,\cdot ]\) is defined as \([ v , \tilde{v} ] = v_{x_{1} x_{1}} \tilde{v}_{x_{2} x_{2}} + v_{x_{2} x_{2}} \tilde{v}_{x_{1} x_{1}} - 2 v_{x_{1} x_{2}} \tilde{v}_{x_{1} x_{2}} \), where \((x_{1}, x_{2}) = x \in \Omega \). The relaxation function k and the exponents \(m (\cdot ) \) and \(p ( \cdot ) \) will be specified later. This type of von Karman equations arises in many applications to the modeling of engineering and physical phenomena such as shells, nonlinear elastic plates, bifurcation theory, and so on.
In the absence of damping, delay, and source effects(\(\alpha = \beta = \gamma =0 \)) in (1.1), many authors [5, 22, 25, 26, 28, 30] have developed decay results by weakening the conditions of the relaxation function k. Munoz Rivera and Menzala [22] showed an exponential decay result when k satisfies
and a polynomial decay result when
The authors of [25, 30] improved those results under the condition
where ζ fulfills \(\zeta (t) >0 \) and \(\zeta '(t) \leq 0 \) on \([0, \infty ) \). Cavalcanti et al. [5] established a general decay result for a wider class of relaxation functions satisfying
where K is an increasing convex function on an interval \([ 0, \varepsilon )\) and satisfies \(K(0) =0 \). Recently, Park [26] established a more general decay result under the very general condition
where K is an increasing convex function satisfying certain conditions, invoked by the pioneering work of Mustafa [23]. In [23], the author introduced condition (1.10) and established an explicit and general decay result for a viscoelastic wave equation when k satisfies (1.10).
On one hand, the study of elliptic, parabolic, and hyperbolic problems with nonlinearities of variable exponent type has been attracting much interest [1, 2, 19, 21, 31]. The nonlinearities of such type describe various physical applications, for example, electrorheological fluids [29], nonlinear elastics [33], non-Newtonian fluids [3], and image precessing [1]. In recent years, some authors studied the following wave equation with such nonlinearities:
In case \(k =0 \) in (1.11), Messaoudi et al. [21] proved the local existence of solution and showed a blow-up result of the solution with negative initial energy when the exponents \(m ( x ) \geq 2 \) and \(p( x ) \geq 2 \) satisfy some hypotheses. Later, in [12], the authors proved the global existence of solution for the same equation by giving some conditions on initial data. Moreover, they showed that the solution decays exponentially when \(m(x) = 2\) and polynomially when \(m(x) \geq 2 \) and \(m_{2} >2 \), where \(m_{2} = \operatorname{ess} \sup_{x \in \Omega } m (x) \), by using an integral inequality introduced by Komornik [17]. In case \(k \neq 0 \) in (1.11), Park and Kang [27] obtained similar results of [21] for the solution with certain positive initial energy. Most recent, Messaoudi et al. [20] established very general decay results when \(m (x) > 1 \) and the relaxation function k satisfies (1.10). Their results generalize and extend the previous results for problem (1.11).
Inspired by these works, in this article, we consider the viscoelastic von Karman system (1.1)–(1.5) with damping, source, and time delay effects of variable-exponent type. Time delay appears in the phenomena depending on some past occurrences as well as on the present state, and may cause instability. We refer to [7, 32] for more applications of time delay and [11, 16, 24] for various decay results of delayed equations. In the absence of memory and time delay(\(k = \beta = 0 \)) in (1.1), Ha and Park [13] proved the global existence of solution and showed exponential or polynomial decay results depending on \(m_{2} \geq 2 \). At this point, it is worth to say that there are no works on the global existence of solution and general decay of the solution for viscoelastic von Karman equations with variable exponent damping and source terms. Due to the presence of source effect, we have some difficulty in deriving desired general decay results. We overcome this by giving some conditions on initial data. Moreover, as far as we know, the global existence and decay of solutions for viscoelastic von Karman equations with delay of variable exponent type have not been considered before. Thus, we intend to discuss the issues for problem (1.1)–(1.5).
Here are the contents of this paper. We give preliminaries in Sect. 2. We show a global existence result in Sect. 3. We establish general decay results for both cases \(1 < m_{1} <2 \) and \(m_{1} > 2 \), where \(m _{1} = \operatorname{ess} \inf_{x \in \Omega } m (x) \).
2 Preliminaries
In this section, we present notations, review necessary materials, give assumptions, and state a local existence result.
We denote by \(\| \cdot \|_{Y} \) the norm of a norm space Y. To simplify notations, we denote \(\| \cdot \|_{L^{s} (\Omega )} \) as \(\| \cdot \|_{s} \) for \(1 \leq s \leq \infty \). We use the letter \(B_{s}\) to denote the embedding constant satisfying
Here, we recall Lebesgue and Sobolev spaces of variable exponents (see e.g. [8, 9, 18]). Let D be a bounded domain of \({\mathbb{R}}^{n}\), \(n \geq 1 \), and \(r : \Omega \to [1, \infty ]\) be a measurable function. The Lebesque space
is a Banach space with respect to the Luxembourg-type norm
It is said that \(r ( \cdot )\) satisfies the log-Hölder continuity condition if
for all \(x,\tilde{ x } \in D \text{ with } | x - \tilde{ x } | < b_{2} \), where \(b_{1} > 0\) and \(0 < b_{2} < 1 \).
Throughout this paper, we let
We remind the following property of von Karman bracket.
Lemma 2.1
([6])
Let \(v_{1}, v_{2}, v_{3} \in H^{2}(\Omega )\). If at least one of them is an element of \(H^{2}_{0}(\Omega )\), then
We give the following assumptions.
\((A_{1})\) The exponents \(p (\cdot ) \) and \(m (\cdot ) \) are continuous functions on Ω̅ satisfying (2.2) and
\((A_{2})\) The coefficients α and β have the relation
\((A_{3})\) \(k : {\mathbb{R}}^{+} \to {\mathbb{R}}^{+} \) is a continuously differentiable function and satisfies
and
where \(K : (0, \infty ) \to (0, \infty )\) is a continuously differentiable function, which is either a linear function or a strictly increasing and strictly convex \(C^{2}\)-function on \((0, \varepsilon ]\), \(\varepsilon \leq k(0)\), \(K(0) = K '(0)=0 \), and ζ is positive, differentiable, and nonincreasing.
Remark 2.1
1. For examples of the function k satisfying \((A_{3})\), we refer to [23].
2. Since K is a strictly increasing and strictly convex \(C^{2}\)-function on \((0, \varepsilon ]\) satisfying \(K(0) = K '(0)=0 \), there exists an extension K̅ of K, which is a strictly increasing and strictly convex \(C^{2}\) function on \((0, \infty )\). We mention [23] for details.
As in [24], we introduce a function y as
Then, problem (1.1)–(1.5) reads as
By virtue of the arguments of [4, 15, 27], we have the following local existence result.
Theorem 2.1
(Local existence)
Under \((A_{1})\), \((A_{2})\), and \((A_{3})\), problem (2.7)–(2.12) has a unique local solution \(( u, y ) \) satisfying
for every \(( u_{0}, u_{1} , y_{0} ) \in H^{2}_{0}(\Omega ) \times L^{2}(\Omega ) \times L^{ m (\cdot ) } ( \Omega \times ( 0, 1) ) \).
3 Global existence
In this section, we derive the global existence of solution to problem (2.7)–(2.12). In the proof of global existence and decay results, we will use the following lemma several times.
Lemma 3.1
Let r be a continuous function on Ω̅ with \(2 \leq r_{1} \leq r(x) \leq r_{2} < \infty \). Then, for \(v \in H^{2}_{0} (\Omega ) \), it holds
Proof
Using (2.1), we have
□
We define the energy E of the solution to (2.7)–(2.12) as
where
and
Let
and
Then we see
and
Lemma 3.2
Let \((A_{1})\), \((A_{2})\), and \((A_{3})\) hold. Then there exists \(C_{0} >0\) satisfying
Proof
From (2.7), (2.8), (2.10), we have
Using (2.9) and the equality \(y (x, 0 , t) = u_{t} (x,t ) \), we get
Using Young’s inequality with \(\frac{ m (x) -1 }{ m (x)} + \frac{ 1 }{ m (x)} =1 \) and the fact that the function \(f (s) = \frac{s-1}{s} \) is increasing for \(s > 0 \), we find
Combining (3.9), (3.10), and (3.11), we obtain
From this, (3.3), and \((A_{2})\), we finish the proof. □
Lemma 3.3
Let \((A_{1})\), \((A_{2})\), \((A_{3})\) hold. If
then
Proof
Suppose that there exists \(0 < t_{0} \leq T \) with \(I ( t_{0} ) \leq 0\). Let \(T_{m}\) be the first time satisfying
Then \(I (t) \geq 0\) for \(t \in [0, T_{m} ] \). Thus, from (3.6), (3.7), and (3.8), we observe
for \(t \in [0, T_{m} ]\). Using (3.1), (3.15), and (3.12), we see
for \(t \in [0, T_{m} ] \), which implies
This contradicts (3.14). □
Remark 3.1
Let the conditions of Lemma 3.3 hold. From the result of Lemma 3.3 and the same argument of (3.15), we also find
From Lemma 3.2 and Lemma 3.3, we have the global existence result.
Theorem 3.1
(Global existence)
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. Then the solution \(( u, y )\) to problem (2.7)–(2.12) is global.
Proof
It suffices to show that \(\|u_{t}\|_{2}^{2} + \| \Delta u \|_{2}^{2} + \|\Delta \chi ( u ) \|_{2}^{2} + \int _{\Omega } \int ^{1}_{0} | y (x, \rho , t ) |^{ m (x) } \,d\rho \,dx \) is bounded independent of t. From (3.7), (3.6), (3.13), (3.8), we have
From this and (3.17), we have
where \(c_{0} = \min \{ \frac{1}{ 4 } , \frac{ \xi \tau }{2} \} \). □
4 General decay results
In this section, we derive general decay results for both cases when \(m_{1} \geq 2\) and when \(1 < m_{1} < 2\) by following the ideas in [20] and [23] with some necessary modification.
First, we let
then, by the arguments of [14, 23], we have the following lemma.
Lemma 4.1
Let \((A_{3})\) be satisfied. Then, for \(v \in L^{2}_{\mathrm{loc}} ( [0, \infty ), L^{2}(\Omega ) ) \), it holds
Now, we define
where \(N > 0 \), \(N_{i} >0 \), \(i = 1, 2 \),
and
Lemma 4.2
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. Then \(L(t)\) is equivalent to \(E(t)\).
Proof
From (3.6), (3.13), and (3.7), we have
Taking \(N > \max \{ N_{1} + N_{2} , \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } , \frac{ 2 }{ \xi } \} \), we finish the proof. □
Lemma 4.3
The function ϒ satisfies
Proof
Using (2.9) and \(y (x, 0, t ) = u_{t}(x,t) \), we get
□
Lemma 4.4
Let \({ g(t) = \int ^{\infty }_{t} k(s) \,ds } \). The following function
satisfies
Proof
Noting \(g'(t) = - k (t) \) and using Young’s inequality, we see
□
From here, c and \(C_{i}\) denote generic constants, \(c_{\delta } > 0 \) denotes a generic constant depending on \(\delta >0 \), and \(c_{\delta } (x) = \frac{ m (x) -1}{ \delta ^{ \frac{1}{ m (x) -1 } } ( m (x))^{ \frac{ m (x)}{ m (x) -1 } } } \). We note that \(c_{\delta } (x) \) is bounded on Ω for fixed \(\delta >0 \), that is, \(| c_{\delta } (x) | \leq c_{\delta } \) for all \(x \in \Omega \).
4.1 General decay for the case \({m_{1} \geq 2 } \)
In this subsection, we derive a general decay result for the case \(m_{1} \geq 2\).
Lemma 4.5
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(m_{1} \geq 2 \), then Φ satisfies
Proof
Using Young’s inequality and (4.1), we have
From (4.5) and (4.6), one sees
Using Young’s inequality with \(\frac{1}{ m (x) } + \frac{ m (x) -1 }{ m (x) } =1 \), (3.1), and (3.17), we find
for \(\delta _{1} >0 \), where \(C_{E(0)} = B_{ m_{1} }^{ m_{1} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{1} -2 }{2} } + B_{ m_{2}}^{ m_{2} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{2} -2 }{2} } \).
Similarly, using (2.5), we have
Combining (4.7), (4.8), (4.9) and taking \(\delta _{1} = \frac{ k_{l} }{ 8 \alpha C_{E(0)}} \), we obtain (4.4). □
Lemma 4.6
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(m_{1} \geq 2 \), then Ψ satisfies
for any \(\delta >0 \).
Proof
Using Young’s inequality, \(k ' = \eta k - k_{\eta } \), (4.1) we get
and
Using (3.17), we infer
here we used the Karman bracket property (see p. 270 in [10])
and
Using (3.1), (3.17), (4.1), we deduce
where \({\overline{C}}_{E(0)} = B_{ 2( p_{1} -1) }^{ 2 ( p_{1} -1) } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ 2 p_{1} - 4 }{2} } + B_{ 2( p_{2} -1 )}^{ 2 ( p_{2} -1) } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ 2 p_{2} - 4 }{2} } \).
Using the similar calculation of (4.8) and Hölder’s inequality, we get
where \(\hat{C}_{E(0)} = B_{ m_{1} }^{ m_{1} } ( \frac{ 8 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{1} -2 }{2} } + B_{ m_{2}}^{ m_{2} } ( \frac{ 8 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{2} -2 }{2} } \).
Similarly, we also have
Applying the estimates of \(J_{i}\) to (4.11), we obtain (4.10). □
Lemma 4.7
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. Moreover, we assume that
If \(m_{1} \geq 2 \), then there exists \(\lambda >0 \) such that
Proof
From \((A_{3}) \), there exists \(t_{ \varepsilon } > 0 \) with \(k( t_{ \varepsilon } ) = \varepsilon \), that is, \(t_{ \varepsilon } = k^{-1}( \varepsilon ) \). We put \(\int _{0}^{ t_{ \varepsilon } } k(s) \,ds = k_{ \varepsilon } \). Using (3.8), (4.4), (4.10), (4.2), and \(k' = \eta k -k_{\eta } \), we have
for \(\lambda >0 \) and \(t \geq t_{ \varepsilon } \). Using estimate (3.16) and taking \(\delta = \frac{ k_{l} }{ 4 N_{2} C_{5}} \), we get
From (4.19), we know
Firstly, we take \(N_{1} > \frac{\lambda }{4} \) large enough to get
and then choose \(N_{2} >0 \) satisfying
Noting \(\frac{ \eta k^{2} (s) }{ k_{\eta } (s)} < k(s) \) and making use of the Lebesgue dominated convergence theorem, we have
Thus, there exists \(0 < \eta _{0} < 1\) satisfying
Secondly, we take \(N > 0\) large enough again to get
and
Thirdly, selecting \(\eta = \frac{1}{ 4 N } < \eta _{0} \), we get
and
here we used (4.25). From (4.22), (4.23), (4.24), (4.26), (4.27), (4.28), (4.29), and (4.30), we get
for \(t \geq t_{ \varepsilon } \). Finally, selecting \(\lambda >0 \) satisfying
we obtain (4.20). □
Lemma 4.8
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(m_{1} \geq 2 \). Then
Proof
and
which gives
□
Theorem 4.1
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(m_{1} \geq 2 \). Then there exist \(c_{i} , \omega _{i} >0 \), \(i =1,2 \), such that, for \(t \geq k^{-1}(\varepsilon ) \),
and
where
Proof
From Lemma 4.5, Lemma 4.6, and Lemma 4.7, the proof is similar to that of [23]. But, for the completeness, we give the proof. Since k and ζ are continuous in t, we have
for some \(a_{1}, a_{2} >0 \), and
From (4.20), (4.35), (3.8), we get
Let
then \(R \sim E \) and
Case 1: K is linear, that is, \(K (s) = a s \) for some \(a > 0\). Put
From (4.37), (2.6), and (3.8), we have
This and the relation \({\mathcal{R}}_{1} (t) \sim E(t) \) prove (4.33).
Case 2: K is nonlinear. For \(t \geq t_{\varepsilon } \), we put
and
From (3.17), (3.8), (4.31), we get
Thus, there exists \(0 < a_{3} < 1 \) satisfying
From (3.8), we know
Using \((A_{3})\), (4.40), the relation \(\overline{ K } ( \varrho t ) \leq \varrho \overline{ K } ( t )\) for \(0 \leq \varrho \leq 1 \) and \(t \in [0, \infty ) \), and Jensen’s inequality, we find
So, we have
Applying this to (4.37), we get
We know that the convex function K̅ satisfies
and
where \(\overline{ K }^{*}\) is the conjugate function of K̅.
Let \(0 < \mu < \min \{ \varepsilon , 2 a_{3} \lambda E(0) \} \) and \({\mathcal{E}}(t) = \frac{E(t)}{E(0)} \). Using \(\overline{K}'(s) >0\), \(\overline{K}''(s) >0 \), \(E'(t) \leq 0 \), \(\overline{K}(0) = \overline{K}'(0) =0\), (4.43), (4.44), and (4.45), we infer
where \(a_{4} = \lambda E(0) - \frac{ \mu }{ 2 a_{3}} > 0 \). Setting
from (4.46) and (4.41), we get
where \(K_{0}(s) = s K'( \mu s) \). Since \({\mathcal{R}}_{2} (t) \sim E(t) \), there exist \(a_{5}, a_{6} >0\) satisfying
Finally, we let
then
Since \(K_{0}\) is an increasing function on \((0,1] \), from (4.48), (4.47), and (4.49), we deduce
where \(\omega _{2} = \frac{ a_{4} a_{5} }{ E(0)} \), and
here K̃ is the function given in (4.34). Because K̃ is strictly decreasing on \((0, \varepsilon ]\), we obtain
□
4.2 General decay for the case \({1 < m_{1} < 2 } \)
In this subsection, we derive a general decay result for the case \(1 < m_{1} < 2\). We let
and
for \(i =1,2 \).
Lemma 4.9
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(1 < m_{1} < 2 \), then Φ satisfies
Proof
We re-estimate \(I_{1}\) and \(I_{2}\) in (4.7) for the case \(1 < m_{1} <2 \). Using Young’s inequality, for \(\delta _{2} >0 \), we have
Noting \(2 m_{1} -2 < 2 m (x) -2 < m (x) < 2 \) for \(x \in \Omega _{1} \) and using Hölder’s inequality with \((2 - m_{1}) + ( m_{1} -1) =1 \), we get
Applying (4.52) to (4.51), we see
As the estimates of (4.8), for \(\delta _{3} >0 \), we find
where \(\tilde{C}_{E(0)} = B_{ m_{-} }^{ m_{-} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{-} -2 }{2} } + B_{ m_{+}}^{ m_{+} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{+} -2 }{2} } \), here
Combining (4.53) and (4.54) and taking \(\delta _{2} = \frac{ k_{l} }{ 16 \alpha B_{2}^{2}} \) and \(\delta _{3} = \frac{ k_{l} }{ 16 \alpha \tilde{C}_{E(0)}}\), we have
Similarly, we have
Adapting (4.55) and (4.56) to (4.7), we obtain (4.50). □
Lemma 4.10
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(1 < m_{1} < 2 \), then Ψ satisfies
for any \(\delta >0 \).
Proof
We re-estimate \(J_{6}\) and \(J_{7}\) in (4.11) for the case \(1 < m_{1} <2 \). Let \(\delta >0 \). Using (4.52), we have
Since \(m(x) \geq 2\) on \(\Omega _{2} \), we can apply the same argument of (4.17) on \(\Omega _{2}\) instead of Ω to obtain
Combining (4.58) and (4.59) and noting that \(\delta c_{\delta }(x) \) is bounded on Ω, we have
Similarly, we find
Substituting (4.12), (4.13), (4.14), (4.15), (4.16), (4.60), and (4.61) into (4.11), we obtain (4.57). □
Lemma 4.11
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold. If \(1 < m_{1} < 2 \), there exists \(\lambda >0 \) such that
Proof
From (3.8), (4.2), (4.50), and (4.57), the proof is similar to that of (4.20) by replacing the constants \(C_{1} \), \(C_{5}\), and \(c_{\delta } (x) \) by \(C_{6} \), \(C_{7}\), and \(\frac{ c_{ \delta } C_{8}}{ \delta } \), respectively, adding
taking \(\delta = \frac{ k_{l} }{ 4 N_{2} C_{7}} \) in (4.21), and using the relation
which is seen from (3.8). □
Lemma 4.12
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(1 < m_{1} < 2 \). Then
Proof
By virtue of (4.62), estimate (4.32) is replaced by
Using (3.8), (4.64), and Young’s inequality with \(( m_{1} -1) + ( 2 - m_{1} ) =1 \), we observe
where \(a_{7} = C_{9}^{ \frac{1}{ m_{1} -1 } } ( \frac{ \lambda }{2} )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \), which yields
for all \(t \geq t_{\varepsilon } \). So, we get
From this and Hölder inequality with \(( 2 - m_{1} ) + ( m_{1} -1 ) =1 \), we obtain
for any \(t_{2} \geq t_{1} \geq 0 \). □
Theorem 4.2
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(1 < m_{1} < 2 \). Then there exist \(c_{i} , \omega _{i} >0 \), \(i =3,4 \), and \(t_{0} > k^{-1}(\varepsilon ) \) satisfying
(i) if K is linear,
(ii) if K is nonlinear,
where
Proof
Owing to (4.62), estimates (4.36) and (4.37) are replaced by
and
for \(t \geq t_{ \varepsilon }\), respectively.
Case 1: K is linear, that is, \(K (s) = a s \) for some \(a > 0\). Due to (4.67), estimate (4.38) is replaced by
We set
where \(a_{9} = C_{9}^{ \frac{1}{ m_{1} -1 } } ( \frac{ \lambda }{2} )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \zeta ( 0 ) \), which satisfies \({\mathcal{R}}_{3}(t) \sim E(t) \). Using (4.68) and the same argument of (4.65), we have
which ensures (4.66).
Case 2: K is nonlinear. We let
Using (4.39) and (4.63), we get
Thus, there exists \(0 < a_{10} < 1 \) satisfying
Using (4.69) and the same argument of (4.42), we can replace estimate (4.42) by
which reads
From this and (4.67), we get
Let \(0 < \mu < \min \{ \varepsilon , 2 a_{10} \lambda E(0) \} \) and \({\mathcal{E}}(t) = \frac{E(t)}{E(0)} \). Using (4.70) and the same argument of (4.46), we obtain
for \(t \geq t_{\varepsilon } \), where \(a_{11} = \lambda E(0) - \frac{ \mu }{ 2 a_{10} } \). Letting
from (4.71) and (4.41), we get
for \(t\geq t_{\varepsilon } \). Define
where \(a_{12} = C_{9}^{ \frac{1}{ m_{1} -1 } } \zeta (0) ( \frac{ a_{11} }{2 E(0) } )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } { \overline{K}}' ( \mu {\mathcal{E}}(0) ) \). Using (4.72) and the same argument of (4.65), we see
Thanks to \(\lim_{ t \to \infty } \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } =0 \), there exists \(t_{0} > t_{\varepsilon } \) such that
which ensures \({\mathcal{R}}_{4} (t) \sim E(t) \sim {\mathcal{L}} (t) \) for \(t > t_{0} \). Moreover, from (4.73) and (4.74), we deduce
and hence
Since
multiplying this by \(( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } )^{ \frac{1}{ m_{1} -1 } } \), we have
for \(t > t_{0} \). So, letting \({\hat{K}}(s) = s^{ \frac{1}{ m_{1} -1 } } K '(s) \), we have
for \(t > t_{0} \), which gives
for \(t > t_{0} \) and some \(\omega _{4} >0 \). □
5 Conclusion
In this paper, we considered a viscoelastic von Karman equation with damping, source, and time delay terms of variable exponent type. Under assumptions \((A_{1}) \), \((A_{2}) \), \((A_{3})\), and (3.12), we showed that the local solution of problem (2.7)–(2.11) is global. Moreover, we established very general decay results of the solution for both cases \(1 < \operatorname{ess} \inf_{x\in \Omega } m(x) <2 \) and \(\operatorname{ess} \inf_{x\in \Omega } m(x) \geq 2 \) by giving additional condition (4.19) on initial data.
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The author is grateful to the anonymous referees for their careful reading and valuable comments.
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This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2020R1I1A3066250).
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Park, SH. General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities. Bound Value Probl 2022, 23 (2022). https://doi.org/10.1186/s13661-022-01602-4
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DOI: https://doi.org/10.1186/s13661-022-01602-4