Abstract
In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalized Young’s inequality are mainly used to prove the decay estimate.
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1 Introduction
In this article, we consider the decay rate results of the solution to a viscoelastic plate equation with past history. Let Ω be a smooth bounded domain of \(\mathbb{R}^{n}\) and u denotes the transverse displacement of waves. Assume \(u_{0}\), \(u_{1}\) are given initial data, then the partial differential equation is governed by the plate equation and is given by:
where ν is the outer unit normal to ∂Ω, b, h are functions (defined later) and ρ is a positive constant (\(\rho >0\) if \(n\geq 2\) and \(0 < \rho \leq \frac{2}{n-2}\) if \(n \geq 3 \)).
Viscoelasticity takes into account time-independent solid behavior (namely, elastic) and time-dependent fluid behavior (namely, viscosity). Some properties of viscoelastic materials are similar to those of elastic solids and some with Newtonian viscous fluids. Due to the significant advancements in the rubber and plastics industries, the importance of material viscoelastic characteristics has been recognized. This type of problem has many applications in several branches of physics, such as quantum mechanics, nuclear physics, supersymmetric field theories, and optics.
According to classical mechanics, any mathematical model that aims to capture the behavior of suspension bridges must have a sufficient number of degrees of freedom and incorporate some nonlinearity. Nonlinearities enable the observation of some hidden phenomena, as would be predicted (see [1] for more details). Logarithmic expression changes the shape of the distribution; it makes the sample less skewed, and in some cases, it reduces the skewness of data. The occurrence of torsional oscillation can be explained by nonlinearity since the behavior of a suspension bridge is nonlinear. Moreover, the amplitude of the oscillation will decrease if we employ logarithmic nonlinearity.
We begin our review with Dafermos’ pioneer paper [2], in which the author presented the following one-dimensional viscoelastic problem:
Dafermos proved that for smooth monotonic decreasing relaxation functions, the solutions go to zero as t tends to infinity, and he also established an existence result. However, no rate of decay was specified. Since then, many authors have tried to prove the local/global existence results and also to calculate the explicit decay result. In [3] the authors considered the viscoelastic wave equation with infinite memory given by the equation:
and established energy-decay results without making any assumption on the boundedness of initial data and any growth constraint on the damping term. Piskin and Polat Pata [4] considered the following nonlinear Petrovsky equation:
They estimated the decay result by using the Nakaos inequality. In [5] the authors considered the following hyperbolic equation with logarithmic nonlinearity:
They established the blow-up of solutions in finite time for negative initial energy under a few assumptions on M. For some more recent works in logarithmic nonlinearity, see [6] and [7]. In [8, 9], the authors considered the problem of the type:
and discussed the decay properties of the semigroup, where A is a strictly positive self-adjoint linear operator, and the memory kernel μ is a decreasing function. They also established necessary and sufficient conditions for the exponential stability under some suitable assumptions on α, β. Giorgi et al. [10] considered the semilinear hyperbolic equation:
where the memory term is bounded and \(k(0)>0\), \(k(\infty )>0\), \(k'(s)\leq 0\). The behavior of solutions over time was examined. Particularly in the autonomous situation, the existence of a global attractor for solutions was achieved. Conti and Pata [11] considered the following semilinear hyperbolic equation:
The authors considered the memory term as a convex decreasing smooth function and g represents a nonlinear term of at most cubic growth satisfying some conditions. Under past history setup, the authors proved the existence of a regular global attractor. Guesmia [12] considered the following problem:
with a class of infinite history kernels satisfying \(\int _{0}^{+\infty }\frac{h(s)}{H^{-1}(-h^{\prime }(s))}\,ds+\sup_{s \in \mathbb{R}\mathbbm{_{+}}}\frac{h(s)}{H^{-1}(-h^{\prime }(s))}<+\infty \), where H is a strictly increasing convex function with \(H(0)\), \(H'(0)=0\) and \(\lim_{t\rightarrow + \infty}H'(t)=+ \infty \). By using the properties of the convex function and Young’s inequality the authors proved the more general decay result. Guesmia and Messaoudi [13] considered:
where \(g_{1}\) and \(g_{2}\) are two positive nonincreasing functions defined on \(\mathbb{R}^{+}\), and \(a_{1}\), \(a_{2}\) are nonnegative bounded function defined on Ω. The authors in [13] proved the general decay result. Later, Mahdi [14] considered the problem (1) with \(h=0\) and established an explicit and general decay rate result. In [15], Al-Mahdi et al. considered the memory-type Timoshenko system with Dirichlet boundary conditions and the system is given by
They established a few decay rate results on the energy function under the unboundedness of the initial data. For more work related to past history problems, refer to [12, 16–20] (and the references therein). Throughout this paper, we consider the following hypothesis:
- (H1):
-
Let \(b:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) be a nonincreasing \(C^{1}\)-function that satisfies:
$$ 0< b(0), 1- \int _{0}^{\infty}b(\tau )\,d\tau =l>0. $$ - (H2):
-
Assume that \(B:(0,\infty )\rightarrow (0,\infty )\) is a \(C^{1}\) function that is linear or a strictly convex \(C^{2}\) function and strictly increasing on \((0,r_{1}]\), where \(r_{1}\leq b(0)\), \(B(0)=B'(0)=0 \), \(\lim_{s\rightarrow \infty} B'(s)=+ \infty \), \(s \mapsto sB'(s)\) and \(s \mapsto s(B')^{-1}(s)\) are convex on \((0,r_{1}]\) and B satisfies
$$ b'(t)\leq -\xi (t)B\bigl(b(t)\bigr), \quad \forall t\geq 0, $$where \(\xi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a \(C^{1}\) nonincreasing positive function with \(\xi (0)>0\).
- (H3):
-
Let \(h :\mathbb{R} \rightarrow \mathbb{R}\) be a nondecreasing continuous function that satisfies (for some \(c_{1}\), \(c_{2}\), ϵ are positive constants):
$$\begin{aligned}& \tilde{h}\bigl( \vert t \vert \bigr) \leq \bigl\vert h(t) \bigr\vert \leq \tilde{h}^{-1}\bigl( \vert t \vert \bigr), \quad\forall \vert t \vert \leq \epsilon , \\& c_{1} \vert t \vert \leq \bigl\vert h(t) \bigr\vert \leq c_{2} \vert t \vert , \quad\forall \vert t \vert \geq \epsilon , \end{aligned}$$where \(\tilde{h}\in C^{1}(\mathbb{R}^{+})\) with \(\tilde{h}(0)=0 \), which is a strictly increasing function. When h̃ is nonlinear, define H to be a strictly convex \(C^{2} \) function in \((0,r_{2}]\), where \(r_{2} > 0 \) such that \(H(t) = \sqrt{t}\tilde{h}(\sqrt{t}) \).
Remark 1
Since B is strictly convex on \((0,r_{1}]\) and \(B(0)=0\), then
In this article, we give certain notations and declare existence results in Sect. 2. In addition, we express a couple of Lemmas in Sect. 2 that will be useful later. We state and establish the decay rate estimate in Sect. 3, as well as present a few examples to demonstrate the decay rate.
2 Preliminaries and existence results
In this section, we review Dafermos’ theory (see [2]) and define the energy functional that is relevant to our problem. We also state a local existence result and a couple of additional Lemmas that will be useful later. We introduce the function η as follows:
then the initial and boundary conditions are obtained as follows:
Observe that, (2) implies
After combining (1) and (2), we obtain the following system
with the following initial and boundary data
The energy functional associated with the system (4) is given by
where \((b\circ \Delta n^{t})(t)=\int _{0}^{\infty} \int _{\Omega}b(s) \vert \Delta \eta ^{t} \vert ^{2} \,d s \,dx\) and \(\|\cdot\|_{2}=\|\cdot\|_{L^{2}(\Omega )}\). Differentiating \(E(t)\) with t and making use of system (4), we obtain
Remark 2
For any \(\epsilon _{0} \in (0,1) \), we obtain that
Theorem 1
Let \((u_{0}(\cdot ,0),u_{1}) \in H_{0}^{2}(\Omega ) \times H_{0}^{2}(\Omega )\), and assume that the hypothesis (H1)–(H3) holds. Then, the problem (1) has weak solution on \([0,T]\).
The proof of the above theorem can be obtained by following similar lines to those given in [21].
Lemma 2.1
(Cf. [14], Lemma 3.1)
There exists a constant \(M > 0\) such that
where \(f_{1}(t)= \int _{0}^{\infty}b(s+t)(1+ \Vert \Delta u_{0}(s) \Vert _{2}^{2})\,ds\).
Lemma 2.2
(Cf. [21], Lemma 9)
Assume that (H1)–(H3) hold, for some \(\epsilon _{0} \in (0,1)\) and \(0< E(0)< d\).Footnote 1Define the functionals
and \(L(t):=mE(t)+\epsilon \psi _{1}(t)+\psi _{2}(t)\), where \(m, \epsilon \geq 0\), then L satisfies the following:
-
(I)
\(L \sim E\) (i.e., \(\alpha _{1}E(t)\leq L(t) \leq \alpha _{2}E(t)\), for some \(\alpha _{1},\alpha _{2}>0\)),
-
(II)
\(L'(t) \leq -mE(t)+c(b \circ \Delta \eta ^{t})(t)+c(b \circ \Delta \eta ^{t})^{\frac{1}{1+\epsilon _{0}}}(t)+c \int _{\Omega } h^{2}(u_{t}) \,dx\).
Lemma 2.3
(Cf. [22], Lemma 4.1)
Let h satisfy \(\mathrm{(H3)}\). Then, the solution of (1) satisfies
where \(G(t):=\frac{1}{ \vert \Omega _{1} \vert }\int _{\Omega} u_{t}h(u_{t}) \,dx \leq -cE'(t)\) and \({\Omega _{1}=\{x \in \Omega : \vert u_{t} \vert \leq \epsilon}\}\) for some \(c,\epsilon >0 \).
Lemma 2.4
(Cf. [17], Lemma 3.3)
Let \(\delta _{0} > 0\) and assume that (H1) and (H2) hold, then we have \(\forall t\geq 0\),
where
3 Decay result
We state and prove the major result in this section. We also provide an example to demonstrate the decay rate result. We introduce a few notations and a function for this purpose:
where \(W(t)=((B^{-1}(t))^{\frac{1}{1+\epsilon _{0}}}+H^{-1})^{-1}\). Further, denote \(\mathcal{S}\) to be the class of functions \(\chi :(0, \infty ) \rightarrow [0, \infty )\) and \(C^{1}\) a function satisfying \(\chi \leq 1\) and \(\chi ' \leq 0\). Also, for fixed \(C_{1},C_{2} >0\), assume that χ satisfies the estimate:
where \(W_{3}^{*}\) (defined in (14)) is the convex conjugate of \(W_{3}\). Let \(\delta _{1}, c>0\) be generic constants, \(q(t)=(t+1)^{-\frac{1}{1+\epsilon _{0}}}\) and
Note: For ϵ small enough with \(0<\epsilon \leq 1\), \(\epsilon W_{4}(s)\in \mathcal{S}\). Hence, the set \(\mathcal{S}\) is nonempty.
Theorem 2
Under the hypothesis of Lemma 2.2, there exists a constant \(C>0\) such that the solution to problem (1) satisfies,
Proof
Using Lemmas (2.2)–(2.4), note that
Assume \(L_{1}(t):=(L+cE)(t)\) and using Lemma (2.1), the above inequality can be written as
Using Remark 1 with \(\theta =\frac{1}{t+1}\) for all \(t>0\), we obtain
Hence,
Denote \(\beta (t):=\max ( \frac{\delta _{0}q(t) \mu (t)}{\xi (t)}, G(t)q(t) )\) and recall \(W(t)=((B^{-1})^{ \frac{1}{1+\epsilon _{0}}}+H^{-1})^{-1}(t)\).
Then (12) becomes, for any \(t\geq 0\) and \(\epsilon _{0} \in (0, 1)\),
Let \(0<\epsilon _{1} <r:=\min \{r_{1},r_{2}\}\), define the functional \(L_{2}\) as
then it is easy to see that \(L_{2} \sim E\), also
Let the convex conjugate of W be denoted by \(W^{*}\) and be defined as
Using the Generalized Young’s inequality, \(W^{*}\) satisfies the following estimate
Therefore, with \(\tilde{A}=W' ( {\epsilon _{1}q(t)}\frac{E(t)}{E(0)} )\) and \(\tilde{B}=W^{-1}(\beta (t))\), (13) leads to
Multiplying the above equation by \(\xi (t)\), we obtain
Define a functional \(L_{3}:=\xi L_{2}+cE \sim E\). Since \(\xi (t) \beta (t) \leq -cE'(t)\) and \(W^{*}(W'(t)) \leq tW'(t)\), we obtain
Consequently, from (8) and choosing \(\epsilon _{1}\) such that \(k:= (\frac{mE(0)}{\epsilon _{1}}-c )>0\), we obtain
Since \(W'_{2}(t)=W'(t)+tW''(t)\), using the property of W, we conclude that \(W_{2}(t), W'_{2}(t)>0\) on \((0,r]\). Making use of (15) with \(\tilde{A}=W' ( {\epsilon _{1}q(t)}\frac{E(t)}{E(0)} )\) and \(\tilde{B}= [\frac{c}{\delta _{1}}f^{\frac{1}{1+\epsilon _{0}}}_{1} ]\) where \(\delta _{1}>0\), we obtain
Now, combining (16) and (17), and choosing \(\delta _{1}\) small enough so that \(k_{1}=(k-\delta _{1})>0\), we have
using the nonincreasing property of \(W_{2}\) and for some \(\gamma _{1}\), \(\gamma _{2}>0\) that satisfies \(\gamma _{1}L_{3}(t) \leq E(t) \leq \gamma _{2}L_{3}(t)\). We have, for some \(\alpha =\frac{\gamma _{1}}{E(0)}>0\),
Assume \(L_{4}(t):=\alpha L_{3}(t)q(t)\), then using (18) we have
where \(C_{1}=\alpha k_{1}>0\) and \(C_{2}=\alpha \delta _{1}>0\). Since, \(L_{3}\sim E\), we have \(L_{4}(t) \leq \alpha _{0}E(t)q(t)\) for some \(\alpha _{0}>0\). In order to establish the decay estimate on \(E(t)\), we will divide the proof into two cases (like in [15]). From the definition of χ, we can estimate \(E(t)\) by considering two cases:
Case I: If \(\alpha _{0}E(t)q(t) \leq 2\frac{W_{4}(t)}{\chi (t)} \), then we obtain
Case II: If \(\alpha _{0}E(t)q(t) \geq 2\frac{W_{4}(t)}{\chi (t)} \), then observe that for any \(0 \leq s \leq t\), we have \(\alpha _{0}q(s)E(s)>2\frac{W_{4}(s)}{\chi (s)}\). Therefore, we obtain
Using Remark 1, \(0 < \chi (t) \leq 1\) and the property of \(W_{2}\), we have for any \(0<\epsilon _{2} \leq 1\), and \(0 \leq s \leq t\),
Using (21), we have for \(0\leq s \leq t\),
Denote the functional \(L_{5}\) as
where \(\epsilon _{2}\) is chosen in such a way that \(L_{5}(0) \leq 1\). Using (8), inequality (22) can be written as,
Since, \(L'_{5}(t)=\epsilon _{2}\chi '(t)L_{4}(t)+\epsilon _{2}\chi (t)L'_{4}(t)- \epsilon _{2}W'_{4}(t)\), using (19) we obtain
and applying (22) to the above inequality, we obtain
From the definition of \(W_{1}\) and \(W_{4}\) we have \(W'_{4}(t)=-C_{1}\xi (t)W_{2}(W_{4}(t))\), and using (9) we see that
therefore (25) leads to,
Integrating the above inequality over \([0,t]\), we observe that
Since \(W_{1}\) is decreasing, \(L_{5}(0) \leq 1\) and \(W_{1}(1)=0\), we have
From (23), we obtain
Similarly, recalling the definition of the functional \(L_{4}\), we obtain
since \(L_{3} \sim E \), for some \(c>0\) we have \(E(t) \leq c L_{3}\), then (27) becomes
and from (20) and (28), we conclude that
Hence, the theorem is proved. □
Remark 3
Assuming that χ satisfies (9) we have established the stability estimate of the energy function. In general, inequality (11) does not lead to the asymptotic stability \(\lim_{t\rightarrow \infty} E(t)=0\) (see [15] for more details). Hence, we have chosen χ in such a way that (9) and (11) give the best possible decay rate for E.
Example 3.1
Here, we present an example that demonstrates the main theorem of this paper. Assume that \(\epsilon _{0}\in (0,1)\), c is a positive generic constant and
where \(0< c_{0}<\frac{2-p}{p-1}\) and p is defined later. Moreover, assume that \(u_{0}\) satisfies
Then, recalling the definitions from (8) and (10), we obtain:
Note that \(q(t)f_{1}^{\frac{1}{(1+\epsilon _{0})}}(t)\sim (t+1)^{ ( \frac{1}{1+\epsilon _{0}} )(\lambda -\frac{1}{p-1})}\).
Now, choose \(\chi (t)=(t+1)^{\gamma}\), where \(\gamma <\min (0,-\frac{1}{p(1+\epsilon _{0})-1} + \frac{1+\lambda -\lambda p}{(p-1)(1+\epsilon _{0})} )\). Then, it is easy to observe that \(\chi (t)\) satisfies (9). Therefore (11) implies
Hence, from the above estimate we conclude that \(\lim_{t\rightarrow \infty}E(t)=0\). From the above estimate on \(E(t)\), we observe that as \(\epsilon _{0}\) converges to zero we obtain a faster rate of convergence when compared to \(\epsilon _{0}\) closer to 1.
Example 3.2
In this example, we assume that \(\epsilon _{0}\in (0,1)\) and b, ξ, B, and H are given by
where \(0< c_{0}<(\kappa -1)(\log ^{\kappa -1}(2))\) and \(c_{1}\) depends on \(c_{0}\) and κ. Moreover, assume that \(u_{0}\) satisfies
It is easy to see that \(q(t)f_{1}^{\frac{1}{(1+\epsilon _{0})}}(t)\sim [(t+1)^{-1} \log ^{ \lambda -(\kappa -1)}(t+2) ]^{ (\frac{1}{1+\epsilon _{0}} )}\).
Now, choose \(\chi (t)= (\log ^{ \frac{(\kappa -1)-\lambda }{2(1+\epsilon _{0})}}(t+2) ) ((t+1)^{ \frac{\epsilon _{0}}{(1+\epsilon _{0})(1+2\epsilon _{0})}} )\), then \(\chi (t)\) satisfies (9). Therefore, (11) implies
Since \(\lambda < \kappa -1\), we conclude that \(\lim_{t\rightarrow \infty}E(t)=0\).
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Notes
where d denotes the depth of the potential well, we refer to [21] for such a construction of d.
References
Gazzola, F.: Mathematical Models for Suspension Bridges: Nonlinear Structural Instability. Modeling, Simulation and Applications, vol. 15. Springer, Berlin (2015)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
Guesmia, A.: New general decay rates of solutions for two viscoelastic wave equations with infinite memory. Math. Model. Anal. 25(3), 351–373 (2020)
Piskin, E., Polat, N.: On the decay of solutions for a nonlinear Petrovsky equation. Math. Sci. Lett. 3(1), 43–47 (2013)
Piskin, E., Irkıl, N.: Blow-up of the solution for hyperbolic type equation with logarithmic nonlinearity. Aligarh Bull. Math. 39(1), 43–53 (2020)
Piskin, E., Calisir, Z.: Decay and blow up at infinite time of solutions for a logarithmic Petrovsky equation. Tbil. Math. J. 13(4), 113–127 (2020)
Irkıl, N., Piskin, E.: Local existence and blow up for p-Laplacian equation with logarithmic nonlinearity. Miskolc Math. Notes 23(1), 231–251 (2022)
Appleby, J.A.D., Fabrizio, M., Lazzari, B., Reynolds, D.W.: On exponential asymptotic stability in linear viscoelasticity. Math. Models Methods Appl. Sci. 16(10), 1677–1694 (2006)
Pata, V.: Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77, 333–360 (2009)
Giorgi, C., Rivera, J.E.M., Pata, V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260(1), 83–99 (2001)
Conti, M., Pata, V.: Weakly dissipative semilinear equations of viscoelasticity. Commun. Pure Appl. Anal. 4(4), 705–720 (2005)
Guesmia, A.: Asymptotic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 382(2), 748–760 (2011)
Guesmia, A., Messaoudi, S.A.: A general decay result for a viscoelastic equation in the presence of past history and finite history memories. Nonlinear Anal. 13, 476–485 (2012)
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
Al-Mahdi, A.M., Al-Gharabli, M.M., Guesmia, A., Messaoudi, S.A.: New decay results for a viscoelastic-type Timoshenko system with infinite memory. Z. Angew. Math. Phys. 72(1), 22 (2021)
Al-Mahdi, A.M., Al-Gharabli, M.M.: New general decay results in an infinite memory viscoelastic problem with nonlinear damping. Bound. Value Probl. 2019, 140 (2019). https://doi.org/10.1186/s13661-019-1253-6
Al-Mahdi, A.M., Al-Gharabli, M.M.: Energy decay in a viscoelastic equation with past hostory and boundary feedback. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2020.1869943
Giorgi, C., Grasselli, M.G., Vittorino, P.: Well-posedness and longtime behavior of the phase-field model with memory in a history space setting. Q. Appl. Math. 59, 701–736 (2001). https://doi.org/10.1090/qam/1866554
Kang, J.-R.: Long-time behavior of a suspension bridge equations with past history. Appl. Math. Comput. 265, 509–519 (2015). https://doi.org/10.1016/j.amc.2015.04.116
Silva, M.A.J., Ma, T.F.: On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type. IMA J. Appl. Math. 78(6), 1130–1146 (2013). https://doi.org/10.1093/imamat/hxs011
Kakumani, B.K., Yadav, S.P.: Global existence and decay estimates for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity. Communicated in Arxiv (https://arxiv.org/pdf/2201.00983.pdf)
Al-Gharabli, M.M., Al-Mahdi, A.M., Messaoudi, S.A.: General and optimal decay result for viscoelstic problem with nonlinear boundary feedback. J. Dyn. Control Syst. 25, 551–572 (2019)
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Kakumani, B.K., Yadav, S.P. Decay estimate in a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. Bound Value Probl 2022, 95 (2022). https://doi.org/10.1186/s13661-022-01674-2
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DOI: https://doi.org/10.1186/s13661-022-01674-2