Abstract
In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies
where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.
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1 Introduction
In this work, we consider the following viscoelastic plate problem with velocity-dependent material density and logarithmic nonlinearity:
equipped with initial and boundary conditions
where Ω is a bounded domain of \(\mathbb{R}^{2}\) with smooth boundary ∂Ω, n is the unit outer normal to ∂Ω, and ρ and α are positive constants. The relaxation function h satisfies the following general condition:
where the functions ξ and H satisfy some conditions specified later. To motivate our work, let us recall some results regarding problems with logarithmic nonlinearity.
1.1 Problems with logarithmic nonlinearity
The logarithmic nonlinearity has many applications in physics such as nuclear physics, optics, and geophysics [1–3]. For the problems with logarithmic nonlinearity, we start with the works of Birula and Mycielski [4] and [5], where they proved that the wave equations with logarithmic nonlinearity have stable and localized solutions. Cazenave and Haraux [6] considered the Cauchy problem
in \(\mathbb{R}^{3}\) and established the existence and uniqueness of the solution. The corresponding one-dimensional problem of (4) was studied by Gorka [1], who established the global existence of weak solutions, provided that \((u_{0},u_{1}) \in H^{1}_{0} \times L^{2}\). Bartkowski and Gorka [2] investigated weak solutions and also proved the existence of classical solutions. Hiramatsu et al. [3] considered the problem
and investigated numerical solutions of this problem without theoretical analysis. Recently, Al-Gharabli et al. [7] considered the problem
and proved existence and decay results of the solutions under the following condition on the relaxation function:
Al-Gharabli et al. [8] considered the problem
and as in [7] proved the existence and decay results for the solutions with imposing the same condition (7). Very recently, Al-Gharabli [9] considered the same problem (6) and established a general decay result for which the relaxation function h satisfies \(h^{\prime}(t)\le-\xi(t) H(h(t))\). For more results on some problems with logarithmic nonlinearity, we refer to the recent works [10–14].
1.2 Problems with infinite memory
Giorgi et al. [15] considered the following semilinear hyperbolic equation with linear memory in a bounded domain \(\varOmega \subset\mathbb{R}^{3}\):
with \(K(0), K(+\infty)>0\) and \(K^{\prime}\le0\) and proved the existence of global attractors for the solutions. Conti and Pata [16] considered the following semilinear hyperbolic equation:
where the memory kernel is a convex decreasing smooth function such that \(K(0)>K(+\infty)>0\), and \(g:\mathbb{R_{+}}\rightarrow\mathbb{R_{+}}\) is a nonlinear term of at most cubic growth satisfying some conditions. They proved the existence of a regular global attractor. Appleby et al. [17] studied the linear integro-differential equation
and established an exponential decay result for strong solutions in a Hilbert space. Pata [18] discussed the decay properties of the semigroup generated by the following equation:
where A is a strictly positive self-adjoint linear operator, \(\alpha>0\), \(\beta\ge0\), and the memory kernel μ is a decreasing function satisfying specific conditions. Subsequently, they established necessary and sufficient conditions for the exponential stability. Guesmia [19] considered the equation
and introduced a new ingenuous approach for proving a more general decay result based on the properties of convex functions and the generalized Young inequality. He used a larger class of infinite history kernels satisfying the condition
with
where \(H:\mathbb{R}_{+}\to\mathbb{R}_{+}\) is an increasing strictly convex function. Using this approach, Guesmia and Messaoudi [20] later considered the equation
in a bounded domain under suitable conditions on \(a_{1}\) and \(a_{2}\) for a wide class of relaxation functions \(h_{1}\) and \(h_{2}\), which are not necessarily decaying polynomially or exponentially, and established a general decay result such that the usual exponential and polynomial decay rates are only particular cases. Messaoudi and Al-Gharabli [7] considered the nonlinear wave equation
in which the relaxation function g satisfies
and they proved a general decay result on the solution energy using an approach different from that introduced by Guesmia [19]. Recently, Al-Mahdi and Al-Gharabli [21] considered the viscoelastic problem
established decay results in which the relaxation function h satisfies
and obtained a better decay rate than that in [19] and [22]. For more results on problems with infinite memory and finite memory, we refer the reader to [23–27]. Motivated by all these works, we intend to establish a three-fold objective:
- (a)
To extend many earlier works for the wave equations such as those discussed in [1, 3, 7, 28–30] to the plate equation with logarithmic nonlinearity.
- (b)
To extend some general decay results, known for the case of finite history, to the case of infinite history where the relaxation function satisfies a wider class of relaxation functions instead of those considered in [7, 8, 12, 19, 21, 29, 31].
- (c)
To drop the boundedness assumptions on the history data considered in many earlier results in [7, 19, 21].
We obtain our results by using the multiplier method with some logarithmic inequalities and some properties of integro-differential equations and inequalities. Our decay result is based on ξ, H, and α. This paper is organized as follows. In Sect. 2, we present some notations, assumptions, and a local and global existence result of our problem. In Sect. 3, we establish some lemmas needed in the proof of our result. Stability results with an example are presented in Sect. 4. Some conclusions are given in Sect. 5.
2 Preliminaries
In this section, we introduce our assumptions and give some useful lemmas. We use c to denote a positive generic constant.
- \((A1)\):
\(h: \mathbb{R}_{+}\to\mathbb{R}_{+}\) is a \(C^{1}\) nonincreasing function satisfying, for some \(\beta_{0} > 0\),
$$ -\beta_{0} h(s) \leq h'(s),\qquad h(t) > 0\quad\text{and}\quad 1- \int_{0}^{+\infty }h(s)\,ds:={\ell} > 0, $$(18)- \((A2)\):
\(H:(0,\infty)\to(0,\infty)\) is a function in \(C^{1}(\mathbb{R}_{+}) \cap C^{2}(\mathbb{R}_{+}^{*})\) that is increasing and strictly convex, with \(H(0)=H^{\prime}(0)=0\) and \(\lim_{s\rightarrow+ \infty} H^{\prime}(s)=+\infty\), \(s\mapsto s H^{\prime}(s) \) and \(s \mapsto s (H^{\prime} )^{-1}(s)\) are convex on \((0,r]\), and there exists a nonincreasing function \(\xi:\mathbb{R}_{+}\to\mathbb{R}_{+}\) such that
$$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\ge0. $$(19)- \((A3)\):
The constant α in (1) is such that \(0<\alpha<\alpha_{0}=\frac{2\pi\ell e^{3} }{c_{p}}\), where \(c_{p}\) is the smallest positive number satisfying \(\| \nabla u\|^{2}_{2} \le c_{p} \| \Delta u\|^{2}_{2}\) for \(u \in H_{0}^{2}(\varOmega)\), where \(\|\cdot\|_{2} =\|\cdot\|_{L^{2}(\varOmega)}\).
Remark 2.1
Assumption \((A3)\) is needed for establishing the local existence of the solutions of problem (1). For more details, we refer to [8].
Remark 2.2
If H is a strictly increasing and strictly convex \(C^{2}\) function on \((0, r]\) with \(H(0) = H'(0) = 0\), then it has an extension H̅ that is strictly increasing and strictly convex \(C^{2}\) function on \((0,+\infty)\). For instance, if \(H(r) = a\), \(H'(r) = b\), and \(H''(r) = C\), we can define H̅ for \(t > r\) by
For simplicity, in the rest of this paper, we use H instead of H̅.
Remark 2.3
Since H is strictly convex on \((0,r]\) and \(H(0)=0\), then
Remark 2.4
The function \(g(s)=\sqrt{\frac{2\pi\ell}{c_{p} s}}-e^{-\frac{3}{2}}\) is a continuous decreasing function on \((0,\infty)\) with
Then there exists a unique \(\alpha_{0} >0\) such that \(g(\alpha_{0})=0\). Moreover,
which implies that the selection of α in \((A3)\) is possible.
The modified energy functional associated with problem (1)–(2) is given by
where
Direct differentiation of (23) using (1)–(2) leads to
Lemma 2.1
([32, 33] (Logarithmic Sobolev inequality))
Letube any function in\(H^{1}_{0}(\varOmega)\), and letabe any positive real number. Then
Corollary 2.1
Letube any function in\(H^{2}_{0}(\varOmega)\), and letabe any positive real number. Then
Lemma 2.2
Let\(\varepsilon_{0} \in(0,1)\). Then there exists\(d_{\varepsilon_{0}}>0\)such that
Proof
Let \(f(s)=s^{ \varepsilon_{0}} (\vert\ln{s}\vert-s )\). Note that f is continuous on \((0,\infty)\), its limit at 0+ is 0+, and its limit at ∞ is −∞. Then f has a maximum \(d_{ \varepsilon_{0}}\) on \((0,\infty)\), so (27) holds. □
2.1 Existence results
In this subsection, we state without proof a local existence result of our problem (1)–(2).
Theorem 2.1
Let\((u_{0},u_{1}) \in H^{2}_{0}(\varOmega) \times H^{2}_{0}(\varOmega)\). Assume that\((A1)\)–\((A3)\)hold and
Then problem (1)–(2) has a weak solution on\([0,T]\).
The proof of Theorem 2.1 can be obtained by following the same arguments as in [8] and adapting the finite history to the infinite case. For the global existence, we have the following:
Theorem 2.2
Assume that\((A1)\)–\((A3)\)hold. Let\((u_{0},u_{1})\in H^{2}_{0}(\varOmega)\times H^{2}_{0}(\varOmega)\)be such that
where\(c^{3}_{*}\)is a positive embedding constant. Then we have:
where
and
The proof of Theorem 2.2 can be obtained by following the same arguments as in [8] by adapting the finite memory to infinite memory.
3 Technical lemmas
In this section, we start by establishing several lemmas needed for the proof of our main result.
Lemma 3.1
There exists a positive constant\(M_{1}\)such that
where\(h_{1}(t):=\int_{0}^{+\infty} h(t+s) (1+\Vert\triangle u_{0}(s)\Vert^{2} ) \,ds\).
Proof
The proof is based on some arguments in [30]. In fact, we have
where \(M_{1} =\max \{2, \frac{4E(0)}{\ell} \}\). □
Lemma 3.2
Assume thathsatisfies\((A1)\). Then, for\(u\in H^{2}_{0}(\varOmega)\),
Proof
The proof can be easily obtained by applying the Cauchy–Schwarz and Poincaré inequalities. □
Lemma 3.3
Assume that\((A1)\)–\((A3)\)and (29) hold. Then the functionals
satisfy, along the solutions of (1)–(2), the following estimates for any\(\delta,\delta_{1},\delta_{2}>0\)and\(\varepsilon_{0} \in (0,1)\):
Proof
The proof of Lemma 3.3 is similar to that in [8] with some adjustments according to the infinite memory case. □
Lemma 3.4
Assume that\((A1)\)–\((A3)\)and (29) hold and let\(\varepsilon_{0} \in(0,1)\). Assume that
Then, forαsmall enough, there exist positive constantsεandNsuch that the functional
satisfies
and, for any\(t\geq0\), there exists a positive constantmsuch that
Proof
For the proof of (38), we refer to [8]. To prove (39), we let \(\int_{0}^{+\infty}h(s)\,ds=: h_{0}\) and using (24), (35), and (36), for \(t \ge0\), we have
Using the definition of \(E(t)\), we obtain, for any \(m>0\),
Using the logarithmic Sobolev inequality (26), we get, for \(0< m<2 \varepsilon\),
At this point, we carefully choose our constant. First, we pick \(0<\varepsilon< h_{0}\). Then for \(\delta_{1}\), \(\delta_{2}\), and δ small enough, we have
and
Then, for N sufficiently large,
Consequently, we get
Finally, we choose m and α small enough so that \(m \le \varepsilon\) (so \(\frac{m \alpha}{4}\le (\varepsilon-\frac{m}{2} )\frac{\alpha}{2}\)),
and
and we get
Using (23), (24), (30), (31), (32), and (37), we have
By choosing a satisfying
we achieve that (28) is satisfied. This selection gives a guarantee that
which completes the proof of (39). □
Remark 3.1
Recalling (23), (24), (30), and (32), we have
which gives
Using (47), for any \(\varepsilon_{0} \in(0,1)\), we obtain that
Lemma 3.5
If\((A1)\)–\((A2)\)are satisfied, then we have, for all\(t > 0\), the estimate
where\(q_{0} > 0\)is small enough, His defined in Remark2.2, and
Proof
To establish (49), we introduce the functional
Then since E is nonincreasing, by (23) we get
Thus \(q_{0}\) can be chosen so small so that, for all \(t > 0\),
Without loss of generality, for all \(t > 0\), we assume that \(\lambda(t)> 0\); otherwise, we get an exponential decay from (39). Using Jensen’s inequality, (2.3), (50), and (53) gives
and hence (49) is established. □
4 Decay result
In this section, we state and prove our main result and provide an example to illustrate our decay results. Let us start introducing some functions and then establishing several lemmas needed for the proof of our main result. As in [30], we introduce the following functions:
where \(G^{-1}(t)= (H^{-1} (t) )^{\frac{1}{1+ \varepsilon_{0}}}\) and \(\varepsilon_{0} \in(0,1)\). Further, we introduce the class S of functions \(\chi: \mathbb{R}_{+} \rightarrow\mathbb{R}_{+}^{*}\) satisfying, for fixed \(c_{1}, c_{2} > 0\) (should be selected carefully in (76)),
and
where \(d >0\), c is a generic positive constant that may change from line to line, \(h_{2}\) and q will be defined later in the proof of our main result, and
Remark 4.1
According to the properties of H introduced in \((A2)\) and the definition of G, we can see that \(G' > 0\) and \(G''> 0 \) on \((0,r]\), \(G_{2}\) is convex increasing and defines a bijection from \(\mathbb{R}_{+}\) to \(\mathbb{R}_{+}\), \(G_{1}\) is decreasing and defines a bijection from \((0,1]\) to \(\mathbb{R}_{+}\), and \(G_{3}\) and \(G_{4}\) are convex increasing functions on \((0,r]\). Then the set S is not empty because it contains \(\chi(s)=\varepsilon G_{5}(s)\) with \(0 <\varepsilon\leq1\) small enough. Indeed, (57) is satisfied (since (55) and (59)).
Theorem 4.1
Assume that\((A1)\)–\((A3)\)and (29) hold. Then for anyχsatisfying (57) and (58) and for any\(\varepsilon_{0} \in(0,1)\), there exists a strictly positive constantCsuch that the solution of (1)–(2) satisfies, for all\(t \geq0\),
where\(G_{5}\)andχare defined in (55) and (57), respectively, andqwill be defined later in the proof.
Proof
Using (39), (48), and (49), for some positive constant m, \(\varepsilon_{0} \in(0,1)\), and any \(t \geq0\), we get
Combining the strict increasing of H and the inequality \(\frac{1}{t+1} < 1\) for \(t > 0\), we obtain
and, then (61) becomes, for any \(t\geq0\) and \(\varepsilon_{0} \in(0,1)\),
For simplicity, we let \(q(t):= q_{0} (t+1 )^{\frac{-1}{1+\varepsilon_{0}}} \) and \(h_{2}(t):=c h_{1}^{\frac{1}{1+\varepsilon_{0}}}(t)\). Then (63) becomes
Further, letting \(G^{-1}(t)= (H^{-1}(t) )^{\frac{1}{1+\varepsilon_{0}}}\) we reduce (64) to
For \(\varepsilon_{1} < r \), let the functional \(\mathcal{F}\) be defined by
which satisfies \(\mathcal{F} \sim E\). Noting that \(G^{\prime\prime}\geq0\), \(q'\leq0\), and \(E'\leq0\), we get
Let \(G^{*}\) be the convex conjugate of G in the sense of Young (see [34]). Then
and \(G^{*}\) satisfies the generalized Young inequality
So, with \(A=G^{\prime} (\varepsilon_{1}\frac{E(t)q(t)}{E(0)} )\) and \(B=G^{-1} (\frac{q(t) \mu(t)}{\xi(t)} )\), using (24) and (66)–(68), we arrive at
Multiplying (69) by \(\xi(t)\), using (50), and the facts that \(\varepsilon_{1}\frac{E(t)q(t)}{E(0)}< r\) and \(G^{\prime} (\varepsilon_{1}\frac{E(t)q(t)}{E(0)} )=G^{\prime } (\varepsilon_{1}\frac{E(t)q(t)}{E(0)} )\), we get
Consequently, recalling the definition of \(G_{2}\) and choosing \(\varepsilon_{1}\) such that \(k=(\frac{m E(0)}{\varepsilon_{1}} -c)>0\), we obtain, for all \(t \in\mathbb{R}_{+}\),
where \(\mathcal{F}_{1}=\xi\mathcal{F}+c E \sim E\) satisfies, for some \(\alpha_{1},\alpha_{2}>0\),
Since \(G^{\prime}_{2}(t)=G^{\prime}(t)+t G^{\prime\prime}(t)\), using the strict convexity of G on \((0,r]\), we find that \(G_{2}^{\prime}(t), G_{2}(t)>0\) on \((0,r]\). Applying the general Young inequality (68) to the last term in (71) with \(A=G^{\prime} (\varepsilon_{1}\frac{E(t)q(t)}{E(0)} )\) and \(B=[\frac{c}{d}h_{2} (t)]\), we have
Now, combining (71) and (73) and choosing d small enough so that \(k_{1}=(k-d)>0\), we arrive at
Using the equivalent property in (72) and the nonincrease of \(G_{2}\), we have, for some \(d_{0}=\frac{\alpha_{1}}{E(0)}>0\),
Letting \(\mathcal{F}_{2}(t):=d_{0} \mathcal{F}_{1}(t)q(t)\) and recalling that \(q'\leq0\), we arrive at,
Then (75) becomes, for some constants \(c_{1}=d_{0} k_{1} > 0\) and \(c_{2}=d_{0} d >0\),
Since \(d_{0}q(t)\) is nonincreasing. Using the equivalent property \(\mathcal{F}_{1}\sim E\) implies that there exists \(b_{0} > 0\) such that \(\mathcal{F}_{2}(t)\geq b_{0} E(t)q(t)\). Since \(\chi(t)\) satisfies (57) and (58), if \(b_{0} q(t) E(t) \leq 2 \frac{G_{5}(t)}{\chi(t)}\), then we get
If \(b_{0} q(t) E(t) > 2 \frac{G_{5}(t)}{\chi(t)}\), then since \(q(t) E(t)\) is a nonincreasing function, for any \(0 \leq s \leq t\), we have \(b_{0} q(s) E(s) > 2 \frac{G_{5}(t)}{\chi(t)}\). Therefore, for any \(0\leq s \leq t\),
Using (21), \(0< \chi\leq1\), and the convexity of \(G_{2}\), we have, for any \(0 <\varepsilon_{2} \leq1\),
Recalling the definition of \(G_{2}\), that is, \(G_{2}(t)=tG'(t)\), (79) becomes
Now, using (78) and the increase of \(G^{\prime}\), for any \(0 \leq s \leq t\), we have
Combining (81) and (80), we arrive at
Now we let
where \(\varepsilon_{2}\) is small enough such that \(\mathcal{F}_{3}(0)\leq 1\). Recalling the definition of \(G_{2}\), (82) becomes, for any \(0 \leq s \leq t\),
Further, we have
Since \(\chi' \leq0\), using (76), for any \(0 \leq s \leq t\) and \(0 < \varepsilon_{2} \leq1\), we obtain
Then, using (58) and (84), we get
From the definitions of \(G_{1}\) and \(G_{5}\) we have
and hence
Now we have
Then, according to (58), we get
Then (87) gives
Thus from (90) and the definitions of \(G_{1}\) and \(G_{2}\) in (55) and (56) we obtain
Integrating (91) over \([0, t]\), we get
Since \(G_{1}\) is decreasing, \(\mathcal{F}_{3}(0)\leq1\), and \(G_{1}(1)=0\), we have
Recalling that \(\mathcal{F}_{3}(t)=\varepsilon_{2} \chi(t)\mathcal{F}_{2}(t)-\varepsilon_{2} G_{5}(t)\), we have
Similarly, recalling that \(\mathcal{F}_{2}(t):=d_{0} \mathcal{F}_{1}(t)q(t)\), we get
Since \(\mathcal{F}_{1}\sim E\), for some \(b>0\), we have \(E(t)\leq b \mathcal{F}_{1}\), which gives
From (77) and (96) we obtain the estimate
where \(c_{3}=\max\{\frac{2}{b_{0}}, \frac{b(1+\varepsilon_{2})}{d_{0}\varepsilon_{2}}\}\). □
In the following example, we illustrate our decay result.
Example 4.2
Let \(h(t)=\frac{a}{(1+t)^{\nu}}\), where \(\nu>1\) and \(0< a<\nu-1\), so that \((A1)\) is satisfied. In this case, \(\xi(t)=\nu a^{\frac{-1}{\nu}}\), \(H(t)=t^{\frac{\nu+1}{\nu}}\), and \(G^{-1}(t)= (H^{-1} (t) )^{\frac{1}{1+ \varepsilon_{0}}}\). Then for any \(\varepsilon_{0} \in(0,1)\), we have \(G(t)=t^{\lambda}\), where \(\lambda:=\frac{(\varepsilon_{0}+1)(\nu+1)}{\nu}>1\). Recall the definitions of the functions \(G_{i}\), \(i=1,\ldots,5\):
where \(a_{i}\), \(i=1,2,3,4,5\), are positive constants depending on a, ν, and \(\varepsilon_{0}\). As in [30], we consider
where \(r < \nu-1\) and \(m_{0} , m_{1} > 0\). Then for some positive constants \(a_{i}\) (\(i = 6,7\)) depending only on a, ν, \(m_{0}\), \(m_{1}\), r, we have
where \(-\nu+1+r<0\). Recalling the definitions of the functions \(h_{1}\), \(h_{2}\), and q, we have
It is clear that condition (58) is satisfied if
Choosing \(\chi(t)=(1+t)^{m}\), where \(m < \min (0, \frac{-1}{\lambda-1}+\frac{(\nu-r)(\nu+1)}{\lambda\nu} )\), we have the following two cases depending on r.
Case 1: If \(0 < r < \nu-1\), then for any \(\varepsilon >0\), there exists \(C_{\varepsilon}>0\) such that we have the following decay rate estimate of E (60):
Case 2: If \(r \leq0\), then for any \(\varepsilon >0\), there exists \(C_{\varepsilon}>0\) such that the decay rate estimate of E (60) is given by:
Thus estimates (102) and (103) give \(\lim_{t\rightarrow +\infty} E(T)=0\).
5 Conclusion
As far as we know, there are no decay results in the literature known for logarithmic plate equation with infinite memory and a wider class of relaxation functions. Our work extends the works for some wave equations treated in the literature to the plate equation with logarithmic nonlinearity. Also, we succeed to extend some general decay results, known for the case of finite history, to the case of infinite history, where the relaxation function satisfies a wider class of relaxation functions. Furthermore, we dropped the boundedness assumption on the history data considered in earlier results in the literature.
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Al-Mahdi, A.M. Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity. Bound Value Probl 2020, 84 (2020). https://doi.org/10.1186/s13661-020-01382-9
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DOI: https://doi.org/10.1186/s13661-020-01382-9