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Existence and general decay estimate for a nonlinear plate problem

Boundary Value Problems20182018:11

https://doi.org/10.1186/s13661-018-0931-0

Received: 6 November 2017

Accepted: 11 January 2018

Published: 18 January 2018

Abstract

In this work, we study a plate equation modelling a suspension bridge with weak damping and hanger restoring force. We prove the well-posedness and establish an explicit and general decay result without putting restrictive growth conditions on the frictional damping term.

Keywords

existencedecay ratesplate equationsuspension bridge

1 Introduction

The study of plate problems has been widely investigated by mathematicians and other scientists. Plate problems have a lot of applications in different areas of science and engineering such as material engineering, mechanical engineering, nuclear physics and optics. In order to describe the structural behaviour and the stability of large structures in our societies, plate models have been extensively used. For instance, the Kirchhoff theory of plates [1] establishes a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. The stability of Kirchhoff plates in the presence of a linear or nonlinear source has been studied by many authors. See, for instance, the results obtained in Komornik [2], Lagnese [3] and Lasiecka [4, 5]. Al-Gharabli and Messaoudi [6] studied the following nonlinear plate problem:
$$ \textstyle\begin{cases} { u_{tt} + \Delta^{2} u + u + h(u_{t}) = ku\ln |u| u,} & \mbox{in } \Omega\times(0,+\infty), \\ { u=\frac{\partial u}{\partial\eta}=0,} &\mbox{on } \partial \Omega\times[0,+\infty), \\ { u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),} &\mbox{in } \Omega, \end{cases} $$
(1.1)
and established decay of solutions. Lu [7] investigated the nonautonomous plate-type evolutionary problem
$$ \textstyle\begin{cases} { u_{tt} + a(x) u_{t} + \Delta^{2} u +\lambda u + f(u) = g(x,t),} &\mbox{in } \Omega\times(0,T), \\ { u=\frac{\partial u}{\partial\eta}=0,} &\mbox{on } \partial \Omega\times[0,T), \\ { u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),} &\mbox{in } \Omega, \end{cases} $$
(1.2)
and proved the existence of a uniform attractor. Ji and Lasiecka [8] considered a semilinear Kirchhoff plate with a nonlinear dissipation acting via moments
$$ \textstyle\begin{cases} { w_{tt}-\gamma\Delta w_{tt} + \Delta^{2} w + f(w)= 0,} &\mbox{in } \Omega\times(0,+\infty), \\ { w=0,\qquad \Delta w=-g(\frac{\partial w_{t}}{\partial\eta}),} &\mbox{in } \partial\Omega\times[0,+\infty), \\ { w(x,0)=w_{0}(x), \qquad w_{t}(x,0)=w_{1}(x),} &\mbox{in } \Omega, \end{cases} $$
(1.3)
and proved that the plate is uniformly stabilizable with uniform energy decay rates with respect to the parameter γ which represents rotational force. Moreover, they showed that as \(\gamma\longrightarrow 0\), the solutions of the Kirchhoff plate equation converge to the solutions of the semilinear Euler-Bernoulli plate, which is also uniformly stable in finite energy norm.
Recently, plate models have also been of great importance in studying the structural behaviour and instability of suspension bridges. The first attempt to model a suspension bridge through a plate is due to Ferrero and Gazzola [9], where the following hyperbolic problem was introduced:
$$ \textstyle\begin{cases} { u_{tt} + \eta u_{t} + \Delta^{2} u + h(x,y,u) = f(x,y,t),} & \mbox{in } \Omega\times(0,T), \\ { u(0,y,t) = u_{xx}(0,y,t) = 0}, &\mbox{for } {(y,t)\in(-\ell ,\ell)\times(0,T),} \\ { u(\pi,y,t) = u_{xx}(\pi,y,t) = 0,} &\mbox{for } {(y,t)\in (-\ell,\ell) \times(0,T),} \\ { u_{yy}(x,\pm\ell,t) + \sigma u_{xx}(x,\pm\ell,t) = 0,} &\mbox{for } { (x,t)\in(0,\pi)\times(0,T),} \\ { u_{yyy}(x,\pm\ell,t) + (2-\sigma)u_{xxy}(x,\pm\ell,t)= 0,} &\mbox{for }{ (x,t)\in(0,\pi) \times(0,T),} \\ { u(x,y,0) = u_{0}(x,y), \qquad u_{t}(x,y,0 )= u_{1}(x, y),} & \mbox{in } \Omega, \end{cases} $$
(1.4)
where \({\Omega=(0,\pi) \times(-\ell,\ell) \subset\mathbb {R}^{2}}\) is a planar rectangular plate, σ is the Poisson ratio, η is the damping coefficient, h is the nonlinear restoring force of the hangers and f is an external force. The well-posedness and long-time behaviour of this problem were proved in [9] under suitable assumptions on h. A quasilinear stationary variant of this equation was as well suggested in [10]. Wang [11] considered the following fourth-order equation:
$$ {u_{tt} + \delta u_{t} + \Delta^{2} u + au = |u|^{m-2} u,} $$
(1.5)
with the same boundary conditions and initial data as in [10]. He proved the existence and uniqueness of local solution and a finite time blow up result. Messaoudi and co-authors [1216] have carried out extensive analysis of the suspension bridge plate model (1.4), where existence, decay and global attractor results have been established. For more related results, see Gazzola and Wang [17], Berchio et al. [18] and the book [19] on mathematical models for suspension bridges by Gazzola.
In this paper, we consider the following fourth-order plate equation:
$$ { u_{tt} + \Delta^{2} u + \beta(t)g(u_{t})+ h(u) =0,} \quad \mbox{in } \Omega\times(0,T) $$
(1.6)
with the same boundary and initial conditions as in (1.4), where g is a nonlinear function to be specified later, β is the damping coefficient and u represents the downward displacement of a vibrating suspension bridge under the effect of weak frictional damping.

The main aim is to discuss the well-posedness of problem (1.6) and the decay rate of the associated energy functional without any restrictive growth condition on the damping term g. For the well-posedness, we reformulate (1.6) into a semigroup setting and apply the semigroup theory (see Pazy [20]). For the decay rate, we exploit some convexity properties used by Mustafa and Messaoudi [21]. The rest of this work is organised as follows. In Section 2, we present preliminary materials which will be helpful in obtaining our results. In Section 3, we discuss the well-posedness of problem (1.6). In Section 4, we study the decay rate of the energy functional associated with problem (1.6).

2 Preliminaries

In this section, we state some preliminary material that will be helpful in achieving our result. We assume that the functions β, g and h satisfy the following assumptions:
(A1): 

\(\beta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) is a nonincreasing differentiable function.

(A2): 
\(h:\mathbb{R}\rightarrow\mathbb{R}\) is a locally Lipschitz nondecreasing function such that \(h(0)=0\), and denote \({H(s) = \int_{0}^{s} h(\tau)\, d \tau}\), which is positive, such that
$${ sh(s)- H(s)\geq0,\quad \forall s\in\mathbb{R}}. $$
(A3): 
\(g:\mathbb{R}\rightarrow\mathbb{R}\) is a locally Lipschitz nondecreasing \(C^{1}\)-function such that there exist \(\epsilon, c_{1}, c_{2}>0\) and an increasing function \({ M\in C^{1} ( [0,+\infty) )}\) with M linear or \(M(0)=M'(0)=0\) is a strictly convex \(C^{2}\)-function on \({[0,\epsilon)}\) such that
$$ \textstyle\begin{cases} c_{1}|s|\leq|g(s)|\leq c_{2}|s|, &\mbox{if } |s|\geq\epsilon, \\ s^{2} + g^{2}(s)\leq M^{-1}(sg(s)),& \mbox{if } |s|\leq\epsilon. \end{cases} $$
(2.1)

Remark 2.1

  1. 1.

    We obtain from assumption (A3) that \(sg(s)>0\) for \(s\neq0\).

     
  2. 2.

    Assumption (A3) with \(\epsilon=1\) was first introduced by Lasiecka and Tataru [22], where decay estimates for a second-order nonlinear wave equation with nonlinear boundary damping were established.

     
  3. 3.

    To achieve our decay result, we borrow the techniques used by Mustafa and Messaoudi in [21] to prove decay estimates for a second-order wave equation with Dirichlet boundary conditions.

     
As in [9], let us introduce the space
$$ {{H_{*}}^{2}(\Omega) := \bigl\{ w\in H^{2}( \Omega): w(0,y)=w(\pi ,y)=0, \forall y\in(-\ell,\ell) \bigr\} }, $$
(2.2)
together with the inner product
$$ {(u,v)_{H_{*}^{2}}= \int_{\Omega}\bigl[\Delta u\Delta v +(1-\sigma) (2u_{xy}v_{xy}- u_{xx}v_{yy}-u_{yy}v_{xx}) \bigr]\, dx\, dy.} $$
(2.3)
For the completeness of the space \({H_{*}}^{2}(\Omega)\), we have the following results by Ferrero and Gazzola [9].

Lemma 2.1

([9])

Assume that \(0<\sigma<\frac{1}{2}\). Then the usual \({H^{2}(\Omega )}\)-norm and the norm defined by \((\cdot,\cdot)_{H_{*}^{2}} = \|\cdot\|_{H_{\ast}^{2}(\Omega)}^{2}\) are equivalent. Moreover, \(H_{*}^{2}(\Omega)\) is a Hilbert space when endowed with the scalar product \((\cdot,\cdot)_{H_{*}^{2}}\).

Theorem 2.1

([9])

Assume that \(0 < \sigma<\frac{1}{2}\) and \({f \in L^{2}(\Omega )}\). Then there exists a unique \({ u \in H_{\ast}^{2}(\Omega)}\) such that
$$ {(u,v)_{H_{*}^{2}} = \int_{\Omega} fv, \quad \forall v\in H_{*}^{2}(\Omega).} $$
(2.4)

Remark 2.2

The function \({u \in H_{\ast}^{2}(\Omega)}\) satisfying (2.4) is called the weak solution of the stationary problem
$$ \textstyle\begin{cases} {\Delta^{2} u = f,} \\ { u(0,y) = u_{xx}(0, y) = u(\pi, y) = u_{xx}(\pi, y) = 0,} \\ { u_{yy}(x, \pm l) + \sigma u_{xx}(x, \pm l)= u_{yyy}(x, \pm l) + (2-\sigma)u_{xxy}(x, \pm l) = 0.} \end{cases} $$
(2.5)

Theorem 2.2

([9])

The weak solution \({ u\in H^{2}_{\ast}(\Omega)}\) of (2.4) is in \(H^{4}(\Omega)\), and there exists a constant \(C=C(l,\sigma )>0\) such that
$$ {\|u\|_{H^{4}(\Omega)}\leq C\|f\|_{L^{2}(\Omega)}.} $$
(2.6)
In addition, if \({u\in C^{4}(\bar{\Omega})}\), then u is a classical solution of (2.5).

Lemma 2.2

([9])

Let \(u\in H_{*}^{2}(\Omega)\) and assume that \({1\leq p < +\infty}\). Then there exists a constant \({C_{*} = C_{*}(\Omega ,p)>0}\) such that
$${\|u\|_{L^{p}(\Omega)}\leq C_{*}\|u\|_{H^{2}_{\ast}(\Omega)}.} $$

3 Well-posedness

In this section, we discuss the well-posedness of problem (1.6). We begin with the definition of a weak solution of problem (1.6).

Definition 3.1

We say that a function
$$ {u\in C \bigl( [0,T], H_{\ast}^{2}(\Omega) \bigr)\cap C^{1} \bigl( [0,T], L^{2}(\Omega) \bigr)} $$
(3.1)
is a weak solution of (1.6) if
$$ \textstyle\begin{cases} { \frac{d}{dt}\int_{\Omega} u_{t}w + (u,w )_{H_{\ast}^{2}(\Omega)} + \beta(t)\int_{\Omega} g(u_{t})w + \int_{\Omega}h(u)w = 0, \quad \forall w\in H_{\ast}^{2}(\Omega),} \\ { u(0) = u_{0}, \qquad u_{t}(0) = u_{1},} \\ \quad \mbox{for a.e } t\in(0,T). \end{cases} $$
(3.2)
Now, we reformulate problem (1.6) into a semigroup setting. Let \(u_{t} = v\), then problem (1.6) becomes
$$ \textstyle\begin{cases} { U_{t} + AU = F(t,U)}, \\ { U(0)= U_{0}, } \end{cases} $$
(3.3)
where
$$ U = \begin{pmatrix} { u}\\ { v} \end{pmatrix} , \qquad AU = \begin{pmatrix} {-v}\\ {\Delta^{2} u } \end{pmatrix} , \qquad F(t,U)= \begin{pmatrix} 0\\ {-h(u)-\beta(t)g(v)} \end{pmatrix} , \qquad U_{0} = \begin{pmatrix} { u_{0}}\\ { u_{1}} \end{pmatrix} . $$
We introduce the Hilbert space
$${\mathcal{H} = H_{*}^{2}(\Omega)\times L^{2}(\Omega)} $$
equipped with the inner product
$$ {(U,V)_{\mathcal{H}}= ( u,\tilde{u} ) _{H_{*}^{2}(\Omega )} + ( v,\tilde{v} ) _{L^{2}(\Omega)},} $$
(3.4)
where
$${U = (u,v )^{T}, \qquad V = (\tilde{u},\tilde{v} )^{T} \in \mathcal{H}.} $$
Next, we consider the following stationary boundary conditions:
$$ \textstyle\begin{cases} { u_{xx}(0, y) = u_{xx}(\pi, y) = 0}, \\ { u_{yy}(x, \pm\ell) + \sigma u_{xx}(x, \pm\ell) = 0,} \\ { u_{yyy}(x,\pm\ell) + (2-\sigma)u_{xxy}(x,\pm\ell)= 0.} \end{cases} $$
(3.5)
The domain of the operator A is defined as
$$ { D(A)= \bigl\lbrace (u,v)\in\mathcal{H}: u\in H^{4}(\Omega)\mbox{ satisfying (3.5) and } v\in H_{*}^{2}(\Omega) \bigr\rbrace .} $$
We have the following existence and uniqueness result for problem (3.3).

Theorem 3.1

Let \({U_{0}\in\mathcal{H}}\) be given. Assume that (A1)-(A3) hold. Then problem (3.3) has a unique global weak solution
$${U\in C \bigl([0,T], \mathcal{H}\bigr) ).} $$

Proof

To achieve this result, we show that the operator A is maximal monotone and F is locally Lipschitz continuous. For the monotonicity and maximality, see [15] for a complete detail proof.

Local lipschitzness: Let \({U,V \in\mathbf{B}_{R}=\{(u,v)\in D(A): \|(u,v)\|_{\mathcal{H}}\leq R\}}\). By using Lemma 2.2, the local lipschitzness of h and g, and the boundedness of β, we get
$$\begin{aligned} { \bigl\Vert F(t,U)-F(t,V) \bigr\Vert ^{2}_{\mathcal{H}}} =& \left \Vert \begin{pmatrix} 0\\ {-h(u)-\beta(t)g(v)} \end{pmatrix} - \begin{pmatrix} 0\\ -{h(\tilde{u})-\beta(t)g(\tilde{v})} \end{pmatrix} \right \Vert ^{2}_{\mathcal{H}} \\ =&{ \int_{\Omega} \bigl\vert \bigl(h(\tilde{u})-h(u)\bigr)+ \beta (t) \bigl(g(\tilde{v})-g(v)\bigr) \bigr\vert ^{2}} \\ \leq&{2C_{R} \Vert u-\tilde{u} \Vert ^{2}_{L^{2}(\Omega)} + 2C_{R}\beta^{2}(0) \Vert v-\tilde{v} \Vert ^{2}_{L^{2}(\Omega)}} \\ \leq&{2C_{R}C_{*} \Vert u-\tilde{u} \Vert ^{2}_{H_{\ast}^{2}(\Omega)} + 2C_{R}\beta^{2}(0) \Vert v-\tilde{v} \Vert ^{2}_{L^{2}(\Omega)}} \\ \leq& {{C_{R}} \bigl( \Vert u-\tilde{u} \Vert ^{2}_{H_{\ast }^{2}(\Omega)} + \Vert v-\tilde{v} \Vert ^{2}_{L^{2}(\Omega)} \bigr)} \\ =& {{C_{R}} \Vert U-V \Vert ^{2}_{\mathcal{H}}.} \end{aligned}$$
(3.6)
So, F is locally Lipschitz. Thus, by the semigroup theory (see Pazy [20]), we obtain a local unique solution
$${U\in C \bigl([0, T_{m}),\mathcal{H} \bigr)\quad \mbox{for some } T_{m}>0.} $$
To obtain a global unique solution, it suffices to show that \({\| U(t)\|_{\mathcal{H}}}\) is bounded independently of t. To this end, we multiply (1.6)1 by \({u_{t}}\) and integrate over Ω to get
$$ {\frac{d}{dt} \biggl( \frac{1}{2} \int_{\Omega}u_{t}^{2} + \frac {1}{2}\| u\|_{H^{2}_{\ast}}^{2} + \int_{\Omega}H(u) \biggr)= -\beta (t) \int_{\Omega}u_{t} g(u_{t}) \leq0. } $$
(3.7)
On the account of assumption (A2) and remark number (2.1)1, we obtain
$$ { \bigl\Vert U(t) \bigr\Vert _{\mathcal{H}}^{2}= \Vert u_{t} \Vert _{L^{2}}^{2} + \Vert u \Vert _{H^{2}_{\ast }}^{2}\leq E(t)\leq E(0),} $$
where
$$ { E(t)=\frac{1}{2} \int_{\Omega}u_{t}^{2} + \frac{1}{2}\| u\| _{H^{2}_{\ast}}^{2} + \int_{\Omega}H(u).} $$
This completes the proof. □

4 Decay of the energy

In this section, we discuss the decay rates of the energy functional associated with problem (1.6). To achieve this, we state and prove several lemmas that will be fundamental in establishing the main result.

4.1 Technical lemmas

The energy functional associated with problem (1.6) is given by
$$ {E(t)= \frac{1}{2} \int_{\Omega}|u_{t}|^{2} + \frac{1}{2} \|u\| _{H_{*}^{2}(\Omega)}^{2} + \int_{\Omega}H(u)}. $$
(4.1)

Lemma 4.1

The energy functional defined in (4.1) satisfies
$$ {\frac{dE(t)}{dt}= -\beta(t) \int_{\Omega }u_{t}g(u_{t}) \leq0.} $$
(4.2)

Proof

Multiplying (1.6) by \(u_{t}\) and integrating over Ω, we obtain
$$ {\frac{d}{dt} \biggl( \frac{1}{2} \int_{\Omega }|u_{t}|^{2} + \frac{1}{2} \|u\|_{H_{*}^{2}(\Omega)}^{2} + \int_{\Omega }H(u) \biggr) + \beta(t) \int_{\Omega}u_{t}g(u_{t}) = 0.} $$
From (A3) we get that \({sg(s)>0}\) for all \({s\neq0}\). Thus, by using (A1), we obtain
$$ {\frac{dE(t)}{dt}= -\beta(t) \int_{\Omega}u_{t}g(u_{t})\leq0.} $$
(4.3)
We note here that the calculations are justified for regular solutions. However, the result in (4.3) remains true for a weak solution by a density argument. □
Define the functional
$$ {F(t) = mE(t)+ \int_{\Omega}uu_{t},} $$
(4.4)
where m is a positive constant to be specified later.

Lemma 4.2

Assume that (A1)-(A3) hold. Then the functional F satisfies, along the solution of (1.6), the estimates
$${F'(t)\leq- E(t)+ C \int_{\Omega} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr)} $$
and
$${F\sim E,} $$
where C is a positive constant.

Proof

By using (1.6), definition (3.2), Lemma 4.1 and exploiting assumptions (A1) and (A2), direct differentiation gives
$$\begin{aligned} { F'(t)} =&{ mE'(t)+ \int_{\Omega}u_{t}^{2} + \int_{\Omega }uu_{tt}} \\ =& {-m\beta(t) \int_{\Omega}u_{t}g(u_{t})+ \int_{\Omega}u_{t}^{2} - \| u \|_{H_{*}^{2}(\Omega)}^{2}- \beta(t) \int_{\Omega}ug(u_{t}) - \int_{\Omega }uh(u)} \\ \leq& { \int_{\Omega}u_{t}^{2} - \frac{1}{2}\|u \|_{H_{*}^{2}(\Omega )}^{2} - \int_{\Omega}H(u)- \beta(t) \int_{\Omega}ug(u_{t})+ \int _{\Omega} \bigl( H(u)-uh(u) \bigr)} \\ \leq&{-E(t) + \frac{3}{2} \int_{\Omega}u_{t}^{2} + \beta(t) \int _{\Omega} \bigl\vert ug(u_{t}) \bigr\vert } \\ \leq& {-E(t)+ C \int_{\Omega} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr).} \end{aligned}$$
(4.5)
Next, we show that \({F\sim E}\). Using Young’s inequality and Lemma 2.2, we have
$$\begin{aligned} { F(t)} \leq&{ mE(t) + \frac{1}{2} \int_{\Omega}u_{t}^{2} + \frac{1}{2}\|u \|_{L^{2}(\Omega)}^{2}} \\ \leq&{ mE(t)+\frac{1}{2} \int_{\Omega}u_{t}^{2} + \frac{C_{*}}{2}\| u \|_{H_{*}^{2}(\Omega)}^{2} \leq\lambda_{2} E(t).} \end{aligned}$$
(4.6)
Also,
$$\begin{aligned} \begin{aligned} { F(t)} &\geq { mE(t) - \frac{1}{2} \int_{\Omega} u_{t}^{2} - \frac{1}{2} \|u\|_{L^{2}(\Omega)}^{2}} \\ &\geq { mE(t)- \frac{1}{2} \int_{\Omega}u_{t}^{2} - \frac {C_{*}}{2} \|u\|_{H_{*}^{2}(\Omega)}^{2}} \\ &= {\frac{(m-1)}{2} \int_{\Omega}u_{t}^{2} + \frac{(m-C_{*})}{2}\| u \|_{H_{*}^{2}(\Omega)}^{2} + m \int_{\Omega}H(u).} \end{aligned} \end{aligned}$$
We choose \({m>0}\) large enough so that \({(m-1), (m-C_{*}) >0 }\) and arrive at
$$ {F(t) \geq\lambda_{1} E(t).} $$
(4.7)
Thus, we get from (4.6) and (4.7) that
$${\lambda_{1} E(t)\leq F(t)\leq\lambda_{2} E(t).} $$
This completes the proof. □
Next, we choose \({0<\epsilon_{1}\leq\epsilon}\) so that
$$ {sg(s)\leq\min\bigl\lbrace \epsilon,M(\epsilon)\bigr\rbrace ,\quad \forall |s|\leq \epsilon_{1}.} $$
(4.8)
Then, for \({|s|\geq\epsilon_{1}}\), the function \({s\longmapsto \frac{|g(s)|}{|s|}}\) is continuous on compact intervals and thus attains its extrema. Thus, it follows from assumption (A3) that
$$ \textstyle\begin{cases} c'_{1}|s|\leq|g(s)|\leq c'_{2}|s|, & \mbox{if } |s|\geq\epsilon_{1}, \\ s^{2} + g^{2}(s)\leq M^{-1}(sg(s)), & \mbox{if } |s|\leq\epsilon_{1}. \end{cases} $$
(4.9)
As in [23], let us partition Ω as follows:
$${\Omega_{1} = \bigl\lbrace (x,y)\in\Omega: |u_{t}|\leq \epsilon_{1} \bigr\rbrace \quad \mbox{and} \quad \Omega_{2} = \bigl\lbrace (x,y)\in\Omega: |u_{t}| > \epsilon_{1} \bigr\rbrace .} $$

Lemma 4.3

The following inequalities hold for any \(\epsilon>0\) along the solution of (1.6):
$$ { \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr)\leq \int _{\Omega_{1}} u_{t}^{2} + C_{*} \epsilon E(t) + C_{\epsilon} \int_{\Omega _{1}} \bigl\vert g(u_{t}) \bigr\vert ^{2}} $$
(4.10)
and
$$ { \int_{\Omega_{2}} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr) \leq C\epsilon E(t)-C_{\epsilon}E'(t),} $$
(4.11)
where \({C_{*}}\) is the embedding constant defined in Lemma 2.2 and \({C_{\epsilon}}\) is a generic positive constant depending on ϵ.

Proof

For the first inequality (4.10), we use Young’s inequality and Lemma 2.2 to get
$$\begin{aligned} { \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr)} \leq&{ \int_{\Omega_{1}} u_{t}^{2} + \epsilon \int_{\Omega_{1}} \vert u \vert ^{2} + C_{\epsilon} \int_{\Omega_{1}} \bigl\vert g(u_{t}) \bigr\vert ^{2}} \\ \leq&{ \int_{\Omega_{1}} u_{t}^{2} + C_{*} \epsilon\|u\|_{H_{*}^{2}(\Omega )}^{2} + C_{\epsilon} \int_{\Omega_{1}} \bigl\vert g(u_{t}) \bigr\vert ^{2}} \\ \leq& { \int_{\Omega_{1}} u_{t}^{2} + C_{*} \epsilon E(t) + C_{\epsilon } \int_{\Omega_{1}} \bigl\vert g(u_{t}) \bigr\vert ^{2}.} \end{aligned}$$
(4.12)
For the second inequality (4.11), we use Lemma 2.2 and Hölder’s inequality to obtain
$$\begin{aligned} \begin{aligned}[b] { \int_{\Omega_{2}} \bigl\vert ug(u_{t}) \bigr\vert } & \leq { \biggl( \int_{\Omega _{2}} \vert u \vert ^{2} \biggr)^{\frac{1}{2}} \biggl( \int_{\Omega_{2}} \bigl\vert g(u_{t}) \bigr\vert ^{2} \biggr)^{\frac{1}{2}} } \\ &\leq{\|u\|_{L^{2}(\Omega)} \biggl( \int_{\Omega_{2}} \bigl\vert g(u_{t}) \bigr\vert ^{2} \biggr)^{\frac{1}{2}}} \\ &\leq{ C_{*}\|u\|_{H_{*}^{2}(\Omega)} \biggl( \int_{\Omega_{2}} \bigl\vert g(u_{t}) \bigr\vert ^{2} \biggr)^{\frac{1}{2}}.} \end{aligned} \end{aligned}$$
(4.13)
Now, from (4.9)1 we observe that
$$ {|s|^{2}\leq c_{1}''sg(s) \quad \mbox{and} \quad \bigl\vert g(s) \bigr\vert ^{2}\leq c_{2}''sg(s) \quad \mbox{for some positive constants } c_{1}'', c_{2}''.} $$
Thus, with this in mind and Young’s inequality, we obtain
$$\begin{aligned} { \int_{\Omega_{2}} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr)} \leq&{ C \int_{\Omega_{2}} u_{t}g(u_{t}) + C \bigl( \|u \|_{H_{*}^{2}(\Omega)}^{2} \bigr) ^{\frac{1}{2}} \biggl( \int_{\Omega_{2}} u_{t}g(u_{t}) \biggr)^{\frac {1}{2}}} \\ \leq&{-C E'(t) + C \bigl( E(t) \bigr) ^{\frac{1}{2}} \bigl(-E'(t) \bigr)^{\frac{1}{2}}} \\ \leq& {-C E'(t) + C \bigl( \epsilon \bigl( E(t) \bigr) -C_{\epsilon}E'(t) \bigr)} \\ =& { C\epsilon E(t) - C_{\epsilon} E'(t).} \end{aligned}$$
(4.14)
 □

Lemma 4.4

For ϵ small enough and two positive constants d, C, the functional defined by
$${L(t)= F_{1}(t)+ C_{\epsilon}E(t), \quad \textit{where } F_{1}(t)= F(t)+ C_{\epsilon}E(t)} $$
satisfies, along the solution of (1.6), the estimate
$$ {L'(t)\leq-dE(t)+ C \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert g(u_{t}) \bigr\vert ^{2} \bigr)} $$
(4.15)
and
$${L\sim E.} $$

Proof

Using Lemmas 4.2 and 4.3, direct computations give
$$\begin{aligned} { F_{1}'(t)} =& { F'(t)+ C_{\epsilon}E'(t)} \\ \leq& {-E(t)+ C \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr) + C \int_{\Omega_{2}} \bigl( u_{t}^{2} + \bigl\vert ug(u_{t}) \bigr\vert \bigr)} \\ \leq& { -E(t)+ C \int_{\Omega_{1}} u_{t}^{2} + CC_{e} \epsilon E(t) + C_{\epsilon} \int_{\Omega_{1}} \bigl\vert g(u_{t}) \bigr\vert ^{2} + C\epsilon E(t)-C_{\epsilon }E'(t)} \\ \leq& {-(1-C\epsilon)E(t)+ C_{\epsilon} \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert g(u_{t}) \bigr\vert ^{2} \bigr)-C_{\epsilon}E'(t).} \end{aligned}$$
That is,
$$ { \bigl(F_{1}(t)+ C_{\epsilon}E(t) \bigr)'\leq -(1-C \epsilon )E(t)+ C_{\epsilon} \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert g(u_{t}) \bigr\vert ^{2} \bigr).} $$
(4.16)
This implies
$$ {L'(t)\leq -(1-C\epsilon)E(t)+ C_{\epsilon} \int_{\Omega _{1}} \bigl( u_{t}^{2} + \bigl\vert g(u_{t}) \bigr\vert ^{2} \bigr).} $$
(4.17)
We then choose ϵ small enough so that \({ (1-C\epsilon)>0}\) and obtain the result. It is easy to see that \({ L\sim E}\) since \({F\sim E}\). This completes the proof. □

4.2 Main decay result

Now, we state and prove our main decay result.

Theorem 4.1

Assume that (A1)-(A3) hold. Then there exist positive constants \(k_{1}\), \(k_{2}\), \(k_{3}\), \(\epsilon_{0}\) such that the solution of (1.6) satisfies
$$ {E(t)\leq k_{3}M_{1}^{-1} \biggl(k_{1} \int_{0}^{t}\beta(s)\, ds + k_{2} \biggr), \quad \forall t\geq0,} $$
(4.18)
where
$$ {M_{1}(t)= \int_{t}^{1}\frac{1}{M_{2}(s)}\, ds,\qquad M_{2}(t)=tM'(\epsilon_{0} t)} $$
(4.19)
and \({M_{1}}\) is strictly decreasing on \((0,1]\) and \({\lim_{t\rightarrow0}M_{1}(t)=+\infty}\).

Proof

We have two cases as follows.

Case I. M is linear on \({(0,\epsilon]}\): Multiplying (4.15) by \(\beta(t)\) and using (4.9)2, we deduce that
$$\begin{aligned} {\beta(t)L'(t)} \leq& {-d\beta(t)E(t)+ C\beta(t) \int _{\Omega_{1}} \bigl( u_{t}^{2} + \bigl|g(u_{t})\bigr|^{2} \bigr)} \\ \leq&{-d\beta(t)E(t)+ C\beta(t) \int_{\Omega _{1}}M^{-1}\bigl(u_{t}g(u_{t}) \bigr)} \\ =&{ -d\beta(t)E(t)+ C\beta(t) \int_{\Omega_{1}}u_{t}g(u_{t})} \\ \leq& {-d\beta(t)E(t)+ C\beta(t) \int_{\Omega}u_{t}g(u_{t})} \\ =& {-d\beta(t)E(t)- CE'(t).} \end{aligned}$$
By using (A1), we obtain
$$ { \bigl(\beta(t)L(t) + CE(t) \bigr)'\leq-d \beta(t)E(t).} $$
(4.20)
Let \({J_{1} = \beta L+ C E} \). Then \({J_{1} \sim E} \) since \({L \sim E} \), and we get from (4.20)
$$ {J_{1}'(t)\leq-k_{1} \beta(t)J_{1}(t).} $$
(4.21)
Simple integration of (4.21) over \({(0,t)}\) and using the fact that \({J_{1}\sim E}\) give
$$ {E(t)\leq k_{2}e^{-k_{1} \int_{0}^{t}\beta(s)\, ds}=c M_{1}^{-1} \biggl(c \int_{0}^{t}\beta(s)\, ds \biggr) .} $$
Case II. M is nonlinear on \({(0,\epsilon]}\). In this case, we consider the functional \({I(t)}\) defined by
$$ {I(t)= \frac{1}{|\Omega_{1}|} \int_{\Omega_{1}} u_{t}g(u_{t}).} $$
We know that M is convex, so \(M^{-1}\) is concave. Thus, Jensen’s inequality yields
$$ {M^{-1}\bigl(I(t)\bigr)\geq\frac{1}{|\Omega_{1}|} \int_{\Omega_{1}}M^{-1}\bigl(u_{t}g(u_{t}) \bigr).} $$
(4.22)
By using (4.9)2, we obtain
$$ {\beta(t) \int_{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert g(u_{t}) \bigr\vert ^{2} \bigr)\leq \beta(t) \int_{\Omega_{1}}M^{-1}\bigl(u_{t}g(u_{t}) \bigr)\leq C\beta(t)M^{-1}\bigl(I(t)\bigr).} $$
(4.23)
We multiply (4.15) by \(\beta(t)\) and use (4.23) to arrive at
$$\begin{aligned} {\beta(t)L'(t)} \leq& {-d\beta(t)E(t)+ C\beta(t) \int _{\Omega_{1}} \bigl( u_{t}^{2} + \bigl\vert g(u_{t}) \bigr\vert ^{2} \bigr)} \\ \leq& {-d\beta(t)E(t)+ C\beta(t)M^{-1}\bigl(I(t) \bigr).} \end{aligned}$$
(4.24)
This implies
$$ {\beta(t)L'(t) + E'(t)\leq-d\beta(t)E(t)+ C \beta(t)M^{-1}\bigl(I(t)\bigr)} $$
since \({E'\leq0}\). Using (A1), we obtain
$$ {R'_{0}(t)\leq-d\beta(t)E(t)+ C\beta(t)M^{-1} \bigl(I(t)\bigr),} $$
where
$$ {R_{0} = \beta L + E\sim E.} $$
(4.25)
Let \({\epsilon_{0}<\epsilon}\), \({C_{0}>0}\) and define the functional
$$ {R_{1}(t)=M' \biggl(\epsilon_{0} \frac{E(t)}{E(0)} \biggr)R_{0}(t)+ C_{0}E(t).} $$
(4.26)
Let us note here that \({E(0)>0}\), otherwise \(E(t)=0\), \(\forall t\in\mathbb{R}^{+}\), and thus the theorem is verified since \({E'(t)\leq0}\). Now, since \({R_{0}\sim E}\) and \({E'\leq0}\), \({M'>0}\) (M is increasing), \({M''>0}\) (M is convex) on \((0,\epsilon]\), then \({R_{1}}\) satisfies the following:
$$ \alpha_{1} R_{1}(t)\leq E(t)\leq \alpha_{2} R_{1}(t) \quad \mbox{for some }\alpha _{1}, \alpha_{2}>0 $$
(4.27)
and it follows from (4.26) that
$$\begin{aligned} { R'_{1}(t)} =& {\epsilon_{0} \biggl(\frac{E'(t)}{E(0)} \biggr) M'' \biggl( \epsilon_{0}\frac{E(t)}{E(0)} \biggr)R_{0}(t)+ M' \biggl(\epsilon_{0}\frac{E(t)}{E(0)} \biggr)R'_{0}(t) + C_{0}E'(t)} \\ \leq& { M' \biggl(\epsilon_{0} \frac{E(t)}{E(0)} \biggr) \bigl[-d\beta(t)E(t)+ C\beta(t)M^{-1} \bigl(I(t)\bigr) \bigr] + C_{0}E'(t)} \\ =& {-d\beta(t)E(t)M' \biggl(\epsilon_{0} \frac{E(t)}{E(0)} \biggr)} \\ &{}+ { C\beta(t)M' \biggl(\epsilon_{0} \frac{E(t)}{E(0)} \biggr)M^{-1}\bigl(I(t)\bigr)+ C_{0}E'(t).} \end{aligned}$$
(4.28)
Now, let \({M^{\ast}}\) be the convex conjugate of M in the sense of Young. Then
$$ {M^{\ast}(s) = s\bigl(M' \bigr)^{-1}(s) - M\bigl(\bigl(M'\bigr)^{-1}(s) \bigr), \quad \mbox{if } s\in\bigl(0, M'(\epsilon)\bigr)} $$
(4.29)
and \(M^{\ast}\) satisfies the generalised Young’s inequality
$$ {XY\leq M^{\ast}(X) + M(Y), \quad \mbox{if } X\in \bigl(0,M'(\epsilon)\bigr), Y\in (0,\epsilon).} $$
(4.30)
Next, we set \({X = M'(\epsilon_{0}\frac{E(t)}{E(0)})}\) and \({Y =M^{-1}(I(t))}\). By using Lemma 4.1, the fact that \(sg(s)\leq\min\lbrace\epsilon,G(\epsilon)\rbrace\), if \({|s|\leq\epsilon_{1}}\) and (4.28)-(4.30), we obtain
$$\begin{aligned} { R'_{1}(t)} \leq& {-d\beta(t)E(t)M' \biggl(\epsilon_{0}\frac {E(t)}{E(0)} \biggr) + C_{0}E'(t)} \\ &{}+ { C\beta(t) \biggl[M^{\ast} \biggl( M' \biggl( \epsilon_{0}\frac {E(t)}{E(0)} \biggr) \biggr) + M\bigl(M^{-1} \bigl(I(t)\bigr)\bigr) \biggr]} \\ =&{ -d\beta(t)E(t)M' \biggl( \epsilon_{0} \frac{E(t)}{E(0)} \biggr)+ C_{0}E'(t)} \\ &{}+{ C\beta(t)M^{\ast} \biggl( M' \biggl( \epsilon_{0}\frac {E(t)}{E(0)} \biggr) \biggr) + C\beta(t)I(t)} \\ =& {-d\beta(t)E(t)M' \biggl( \epsilon_{0} \frac{E(t)}{E(0)} \biggr)+ C_{0}E'(t)} \\ &{}+ { C\epsilon_{0}\beta(t) \biggl(\frac{E(t)}{E(0)} \biggr) M' \biggl(\epsilon_{0}\frac{E(t)}{E(0)} \biggr) -C \beta(t)M \biggl( \epsilon _{0}\frac{E(t)}{E(0)} \biggr) + C \beta(t)I(t)} \\ \leq& {-E(0)d\beta(t) \biggl(\frac{E(t)}{E(0)} \biggr) M' \biggl(\epsilon_{0}\frac{E(t)}{E(0)} \biggr) + C\epsilon_{0} \beta(t) \biggl(\frac{E(t)}{E(0)} \biggr) M' \biggl( \epsilon_{0}\frac{E(t)}{E(0)} \biggr)} \\ &{}+ { C\beta(t) \int_{\Omega} u_{t}g(u_{t})+ C_{0}E'(t)} \\ \leq& {-E(0)d\beta(t) \biggl(\frac{E(t)}{E(0)} \biggr) M' \biggl(\epsilon_{0}\frac{E(t)}{E(0)} \biggr) + C\epsilon_{0} \beta(t) \biggl( \frac{E(t)}{E(0)} \biggr) M' \biggl( \epsilon_{0}\frac{E(t)}{E(0)} \biggr)} \\ &{}- { CE'(t)+ C_{0}E'(t).} \end{aligned}$$
We choose \({C_{0}}\) large enough and \(\epsilon_{0}\) small enough such that
$${C-C_{0} < 0, \qquad E(0)d - C\epsilon_{0}>0,} $$
and arrive at
$$ {R'_{1}(t)\leq- k\beta(t) \biggl( \frac{E(t)}{E(0)} \biggr) M' \biggl(\epsilon_{0} \frac{E(t)}{E(0)} \biggr)= -k\beta(t)M_{2} \biggl( \epsilon _{0}\frac{E(t)}{E(0)} \biggr),} $$
(4.31)
where \({M_{2}(t)=tM'(\epsilon_{0}t).}\) We have that
$$ {M'_{2}(t)= M'(\epsilon_{0}t) + \epsilon_{0} tM''(\epsilon_{0}t).} $$
Thus, using the strict convexity of M on \({(0,\epsilon]}\), we get that \({M_{2}, M'_{2}>0}\) on \((0,1]\). It follows from (4.27) and (4.31) that the functional
$${R_{2}(t)=\alpha_{1}\frac{R_{1}(t)}{E(0)}} $$
satisfies
$$ {R_{2}\sim E} $$
(4.32)
and
$$ {R_{2}'(t)\leq-k_{1} \beta(t)M_{2}\bigl(R_{2}(t)\bigr) \quad \mbox{for some } k_{1}>0.} $$
(4.33)
Inequality (4.33) implies that
$${\bigl(M_{1}\bigl(R_{2}(t)\bigr)\bigr)'\geq k_{1}\beta(t),} $$
where
$$ {M_{1}(\tau)= \int_{\tau}^{1}\frac{1}{M_{2}(s)}\, ds,\quad \tau \in(0,1].} $$
Thus, integrating (4.33) over \((0,t)\) and noting that \({ M_{1}}\) is strictly decreasing on \((0,1]\) give
$$ {R_{2}(t)\leq M_{1}^{-1} \biggl(k_{1} \int_{0}^{t} \beta(s)\, ds + k_{2} \biggr) \quad \mbox{for some } k_{2}>0.} $$
(4.34)
Combining (4.32) and (4.34), we get the result. This completes the proof. □

5 Examples

In this section, we illustrate our result with some examples. As in [21], let \({g_{0}\in C^{2} ( [0,+\infty) )}\) be a strictly increasing function such that \({g_{0}(0)=0}\), and for some positive constants \(c_{1}\), \(c_{2}\) and ϵ, the function g satisfies
$$ \textstyle\begin{cases} c_{1}|s|\leq|g(s)|\leq c_{2}|s|,& \forall |s|\geq\epsilon, \\ g_{0}(|s|)\leq|g(s)|\leq g_{0}^{-1}(|s|),& \forall |s|\leq\epsilon. \end{cases} $$
(5.1)
Define the function
$$ {M(s)= \biggl( \sqrt{\frac{s}{2}} \biggr)g_{0} \biggl(\sqrt{ \frac {s}{2}} \biggr).} $$
(5.2)
Then M is a \(C^{2}\)-strictly convex function on \((0,\epsilon]\) when \({g_{0}}\) is nonlinear and thus satisfies assumption (A3). We give some examples of \({g_{0}}\) such that g satisfies (5.1) near 0.
  1. (1)
    Let \({g_{0}(s)= ks}\), where k is a positive constant, then \({M(s)= ks}\) satisfies (A3), and we get
    $${E(t)\leq ke^{-k_{1} \int_{0}^{t}\beta(s)\, ds},\quad \forall t\geq0.} $$
     
  2. (2)
    Let \({g_{0}(s) = \frac{1}{s}e^{-\frac{1}{s^{2}}}}\), then \({M(s)= e^{-\frac{2}{s}}}\) satisfies (A3) near 0 and
    $${E(t)\leq k \biggl(\ln \biggl(k_{1} \int_{0}^{t} \beta(s)\, ds+ k_{2} \biggr) \biggr)^{-1},\quad \forall t\geq0.} $$
     
  3. (3)
    Let \({g_{0}(s) = e^{-\frac{1}{s}}}\), then \({M(s)= \sqrt{\frac{s}{2}} e^{-\sqrt{\frac{2}{s}}}}\) satisfies (A3) near 0, and we obtain
    $${E(t)\leq k \biggl(\ln \biggl(k_{1} \int_{0}^{t} \beta(s)\, ds + k_{2} \biggr) \biggr)^{-2},\quad \forall t\geq0.} $$
     

6 Conclusion

This paper has been able to establish the well-posedness and decay estimate for a nonlinear plate equation with a partially hinged boundary condition. We also illustrated our result with some examples. This result is new for these types of problems, and it generalises many related problems in the literature.

Declarations

Acknowledgements

The author would like to appreciate anonymous refferees for their careful reading and very useful comments. The continuous support of the University of Hafr Al Batin is also appreciated.

Funding

There is no funding budget for this research.

Authors’ contributions

All authors read and approved the final manuscript.

Competing interests

The author declares that there are no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia

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