Global attractor for a nonlinear Timoshenko equation with source terms

Abstract

We study weak solutions of the Timoshenko equation in a bounded domain. We consider a nonlinear dissipation and a nonlinear source term. We obtain boundedness of the solutions as well as their asymptotic behavior. In particular, the source term does not produce a blowup, and the global attractor is the set of all equilibria.

Introduction

In this work, we shall study the dynamics of the following equation:

$$u_{\mathit{\text{tt}}}+{\mathrm{\Delta }}^{2}u-M(\parallel \nabla u{\parallel }_2^2)\mathrm{\Delta }u+g\left(u_t\right)=f\left(u\right)\text{in}\mathrm{\Omega },$$

(1)

with one set of the following boundary conditions:

$$u=0\text{and}\mathrm{\Delta }u=0\text{on}\partial \mathrm{\Omega }$$

or

$$u=0\text{and}\frac{\partial u}{\partial \nu }=0\text{on}\partial \mathrm{\Omega },$$

and the following initial conditions:

$$u(x,0)=u_0,u_t(x,0)=v_0,x\in \mathrm{\Omega }.$$

Here, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain with a sufficiently smooth boundary, ∥·∥₂ is the norm in L²(Ω), and the nonlinearities considered are defined by:

$$M\left(s^2\right)=\alpha +\beta s^{2\gamma },\alpha \ge 0,\beta \ge 0,\alpha +\beta >0,\gamma \ge 1,$$

(2)

$$g\left(u_t\right)=\delta u_t|u_t{|}^{\lambda -2},\delta >0,\lambda \ge 2,$$

(3)

and

$$f\left(u\right)=\mu u|u{|}^{r-2},\mu >0,r>2.$$

(4)

For n = 1, Timoshenko equation is an approximate model describing the transversal motion of a rod. See the work of Antman [1] for a general and rigorous framework of models in the theory of elasticity, in particular, of Equation 1. Here, we are interested in the qualitative behavior of solutions of the Timoshenko equation for any n. The dynamics of second-order equations in time has been widely studied by Alves and Cavalcanti [2], Barbu et al. [3], Cavalcanti et al. [4–9], Rammaha and Sakuntasathien [10, 11], Todorova and Vitillaro [12, 13]. There are a number of papers studying the dynamics of Equation 1, when $\mathit{\text{sf}}\left(s\right)\le 0,s\in \mathbb{R}$; see for instance the books of Hale [14] and Haraux [15] and references therein. For a destabilizing source term, $\mathit{\text{sf}}\left(s\right)>0,s\in \mathbb{R}\setminus \left\{0\right\}$, there are several results studying the effect of this force in nonlinear wave equations; see the papers of Payne and Sattinger [16], Georgiev and Todorova [17], Ikehata [18], and Esquivel-Avila [19, 20]. For the undamped Timoshenko equation, Bainov and Minchev [21] gave sufficient conditions for the nonexistence of smooth solutions of (1), with negative initial energy, and gave an upper bound of the maximal time of existence. For positive and sufficiently small initial energy, blowup and globality properties are characterized in the study of Esquivel-Avila [22]. For damped Timoshenko equation, see another study of Esquivel-Avila [23]; we proved blowup in finite time, globality and unboundedness, globality and convergence to the equilibria, and rates of such convergence for the zero equilibrium. All these results were obtained by means of the potential well theory under the assumption that r ≥ 2(γ + 1). To the knowledge of the author, the behavior of the solutions is still unknown when 2 < r < 2(γ + 1). Here, we prove that, in this case, there is no blowup; all of the solutions are global, are uniformly bounded, and converge to the equilibria set.

Preliminaries

We present an existent, unique, and continued theorem for Equation 1 (see [23]).

Theorem 1

Assume that r > 2 and r ≤ 2(n − 2)/(n − 4) if n ≥ 5. For every initial data (u₀,v₀) ∈ H ≡ B × L²(Ω), where B is defined either by$B\equiv H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)$or by$B\equiv H_0^2\left(\mathrm{\Omega }\right)$, there exists a unique (local) weak solution (u(t),v(t)) of problem (1), that is,

$$\begin{array}{ll}\frac{d}{\mathit{\text{dt}}}{\left(v\right(t),w)}_2& +{\left(\mathrm{\Delta }u\right(t),\mathrm{\Delta }w)}_2+M(\parallel \nabla u(t\left){\parallel }_2^2\right){(\nabla u(t),\nabla w)}_2\\ +{\left(g\right(v\left(t\right)),w)}_2={\left(f\right(u\left(t\right)),w)}_2,\end{array}$$

(5)

almost everywhere (a.e.) in (0,T) and for every w ∈ B ∩ L^λ(Ω), such that

$$\begin{array}{l}u\in C\left(\right[0,T);B)\cap C^1\left(\right[0,T);L^2(\mathrm{\Omega }\left)\right),\\ v\equiv u_t\in L^{\lambda }\left(\right(0,T)\times \mathrm{\Omega }).\end{array}$$

Here, (·,·)₂ denotes the inner product in L²(Ω).

The following energy equation holds:

$$E_0=E\left(t\right)+{\int }_0^t\delta \parallel v\left(\tau \right){\parallel }_{\lambda }^{\lambda }d\tau ,$$

(6)

where

$$E\left(t\right)\equiv E\left(u\right(t),v(t\left)\right)\equiv \frac{1}{2}\parallel v\left(t\right){\parallel }_2^2+J\left(u\right(t\left)\right)$$

(7)

and

$$J\left(u\right)\equiv \frac{1}{2}a\left(u\right)+\frac{1}{2(\gamma +1)}c\left(u\right)-\frac{1}{r}b\left(u\right),$$

(8)

with

$$\begin{array}{l}a\left(u\right)\equiv \parallel u{\parallel }_B^2\equiv \parallel \mathrm{\Delta }u{\parallel }_2^2+\alpha \parallel \nabla u{\parallel }_2^2,b\left(u\right)\equiv \mu \parallel u{\parallel }_r^r,\\ c\left(u\right)\equiv \beta \parallel \nabla u{\parallel }_2^{2(\gamma +1)}.\end{array}$$

(9)

Here, E₀ ≡ E(u₀,v₀) is the initial energy, and ∥·∥ _q  denotes the norm in the L^q(Ω) space.

If the maximal time of existence T _M  < ∞, then (u(t),v(t)) → ∞ as t ↗ T _M , in the norm of H:

$$\parallel (u,v){\parallel }_H^2\equiv \parallel u{\parallel }_B^2+\parallel v{\parallel }_2^2.$$

(10)

In that case, from (6) to (9), ∥u(t)∥ _r  → ∞ as t ↗ T _M .

We define the set of equilibria of Equation 1 by:

$$\mathcal{E}\equiv \{u_e\in B:{\mathrm{\Delta }}^2{u}_e-M(\parallel u{\parallel }_2^2)\mathrm{\Delta }{u}_e=f(u_e\left)\right\}.$$

(11)

We notice that, in particular, $0\in \mathcal{E}$.

Main result

In this section, we prove that all of the solutions are global and uniformly bounded and that the global attractor is $\mathcal{E}$.

Theorem 2

Let (u(t),v(t)) be a solution of problem (1), given by Theorem 1. Assume that r < 2(γ + 1) and r ≤ 2n/(n − 2) if n ≥ 3. Then, (u(t),v(t)) is global and uniformly bounded, and$\left(u\right(t),v(t\left)\right)\to {\mathcal{E}}_{\infty }$is strongly in H as t → ∞, where${\mathcal{E}}_{\infty }\equiv \left\{\right(u_e,0):u_e\in \mathcal{E},J(u_e)=E_{\infty }\equiv \underset{t\to \infty }{lim}E(u\left(t\right),v\left(t\right)\left)\right\}.$

Proof

Since the proof is long, we shall divide it into five steps as follows. First, we prove that the solution is global and bounded. Next, we show that this implies weak convergence to the equilibria set. In order to conclude strong convergence, we have to prove that the orbit is precompact in the phase space. In order to do that, we show that the solution is uniformly continuous. We then prove the precompactness of the orbit. □

Globality and boundedness

Notice that, from the continuous injection $H_0^1\left(\mathrm{\Omega }\right)\hookrightarrow L^r\left(\mathrm{\Omega }\right)$,

$$\begin{array}{ll}c{\left(u\right)}^{\frac{1}{2(\gamma +1)}}& ={\beta }^{\frac{1}{2(\gamma +1)}}\parallel \nabla u{\parallel }_2\ge {\beta }^{\frac{1}{2(\gamma +1)}}C\left(\mathrm{\Omega }\right)\parallel u{\parallel }_r\\ ={\beta }^{\frac{1}{2(\gamma +1)}}C\left(\mathrm{\Omega }\right){\mu }^{-\frac{1}{r}}b{\left(u\right)}^{\frac{1}{r}}\equiv {\kappa }^{\frac{1}{2(\gamma +1)}}b{\left(u\right)}^{\frac{1}{r}},\end{array}$$

and C(Ω) > 0 is a Sobolev constant. Then, along the solution, for any t ≥ 0,

$$\frac{1}{2(\gamma +1)}c(u(t))\ge \frac{\kappa }{2(\gamma +1)}b{\left(u\right(t\left)\right)}^{\frac{2(\gamma +1)}{r}}$$

and

$$\begin{array}{l}\frac{1}{2(\gamma +1)}c\left(u\right(t\left)\right)-\frac{1}{r}b\left(u\right(t\left)\right)\ge \frac{\kappa }{2(\gamma +1)}b{\left(u\right(t\left)\right)}^{\frac{2(\gamma +1)}{r}}\\ \times \left(1-\nu b{\left(u\right(t\left)\right)}^{-\left(\frac{2(\gamma +1)-r}{r}\right)}\right),\end{array}$$

(12)

where $\kappa \nu \equiv \frac{2(\gamma +1)}{r}.$ Here, we have either

$$(i)b(u(t))\le {\nu }^{\frac{r}{2(\gamma +1)-r}}\text{or}(\mathit{\text{ii}})b(u(t))\ge {\nu }^{\frac{r}{2(\gamma +1)-r}}.$$

We use the energy equation in both cases. For the first one, we get the following from (12):

$$\begin{array}{l}\parallel \left(u\right(t),v(t\left)\right){\parallel }_H^2\le \parallel v\left(t\right){\parallel }_2^2+a\left(u\right(t\left)\right)+\frac{1}{\gamma +1}c\left(u\right(t\left)\right)\\ \le 2E_0+\frac{2}{r}{\nu }^{\frac{r}{2(\gamma +1)-r}}.\end{array}$$

In the second case,

$$\parallel \left(u\right(t),v(t\left)\right){\parallel }_H^2=\parallel v\left(t\right){\parallel }_2^2+a\left(u\right(t\left)\right)\le 2E\left(t\right)\le 2E_0.$$

Therefore, for any t ≥ 0,

$$\parallel \left(u\right(t),v(t\left)\right){\parallel }_H^2\le 2E_0+\frac{2}{r}{\nu }^{\frac{r}{2(\gamma +1)-r}}\equiv {Ê}_0.$$

(13)

Weak convergence to $\mathcal{E}$

Since the solution is global and uniformly bounded in the norm of H, there exists a sequence, {t _n }, such that if n → ∞, then t _n  → ∞, and $\left(u\right(t_n),v(t_n\left)\right)\to (û,\widehat{v})$ weakly in H. Moreover, $b\left(u\right(t_n\left)\right)\to b\left(û\right)$, because of the compact injection B ↪ L^r(Ω). On the other hand, the energy is uniformly bounded and nonincreasing; consequently,

$$-\infty <E_{\infty }\equiv \underset{t\to \infty }{\text{lim}}E\left(t\right)=\underset{t\ge 0}{inf}E\left(t\right)\le E\left(t\right)\le E_0<\infty .$$

(14)

From the energy equation and the continuous injection L^λ(Ω) ↪ L²(Ω), this implies that

$$\underset{t\to \infty }{\text{lim}}{\int }_t^{t+1}\parallel v(\tau ){\parallel }_2^{\lambda }d\tau =0.$$

In particular, for any sequence {s _n } such that s _n  → ∞ as n → ∞,

$$\underset{n\to \infty }{\text{lim}}{\int }_0^1{h}_n(\tau )d\tau =0,$$

where $h_n\left(\tau \right)\equiv \parallel v(s_n+\tau ){\parallel }_2^{\lambda }$, for τ ∈ [0,1]. By Fatou Lemma,

$$\underset{n\to \infty }{\text{lim\hspace{0.17em}inf}}\parallel v(s_n+\tau ){\parallel }_2^{\lambda }=\underset{n\to \infty }{\text{lim\hspace{0.17em}inf}}{h}_n(\tau )=0,$$

for a.e. τ ∈ [0,1], and by the weak convergence to $\widehat{v}$,

$$\parallel \widehat{v}{\parallel }_2\le \underset{n\to \infty }{\text{lim\hspace{0.17em}inf}}\parallel v(t_n){\parallel }_2=0,$$

where we choose {s _n } such that t _n  = s _n  + τ₀, for some τ₀ ∈ [0,1]. Then, the weak limit set of the orbit is such that

$${\omega }_w(u_0,v_0)=\left\{\right(û,0):(u\left(t_n\right),v\left(t_n\right))\to (û,0),\mathrm{weakly}\mathit{\text{in}}H\}.$$

The weak limit set is positive invariant (see Ball [24]), that is,

$$\left(u\right(0),v(0\left)\right)\in {\omega }_w(u_0,v_0)\Rightarrow \left(u\right(t),v(t\left)\right)\in {\omega }_w(u_0,v_0),\forall t>0.$$

Consequently ${\omega }_w(u_0,v_0)\subset \mathcal{E}$, that is, there exists some $u_e\in \mathcal{E}$, such that, along a sequence of times, the solution converges weakly to (u _e ,0).

Strong convergence to $\mathcal{E}$

If the convergence is strong in H,

$$\underset{n\to \infty }{\text{lim}}\parallel (u(t_n)-u_e,v(t_n)){\parallel }_H=0,$$

then

$$J(u_e)=\underset{n\to \infty }{\text{lim}}J(u(t_n)),$$

and by (14),

$$E_{\infty }\equiv \underset{n\to \infty }{\text{lim}}E(t_n)=\underset{t\to \infty }{\text{lim}}J(u(t_n)).$$

Consequently,

$$E_{\infty }\equiv \underset{t\to \infty }{\text{lim}}E(t)=J(u_e),$$

and the assertion of the theorem holds.

Now, strong convergence follows if the orbit

$${\{(u(t),v(t))\}}_{t\ge 0}$$

(15)

is a precompact subset of H.

To show this, we shall use a technique due to Haraux [15], and we shall extend it to handle the nonlinearities of Equation 1 as follows.

Uniform continuity

We shall prove that t ↦ (u(t),v(t)) ∈ H is uniformly continuous. In order to do that, we define, for every ε > 0 and t ≥ 0,

$$\begin{array}{ll}{u}_{\epsilon }(t)& \equiv u(t+\epsilon )-u(t),v_{\epsilon }(t)\equiv v(t+\epsilon )-v(t),\\ 2w_{\epsilon }(t)\equiv \parallel (u_{\epsilon }(t),v_{\epsilon }(t){\parallel }_H^2.\end{array}$$

Hence, the uniform continuity of the solution holds if for any η > 0, there exists ε(η) > 0, such that

$$w_{\epsilon }(t)\le \eta ,$$

(16)

for every t ≥ 0, and ε ∈ (0,ε(η)). To get that estimate, we need the energy equation for (u _ε (t),v _ε (t)). Then, from (5), we obtain

$$w_{\epsilon }(0)=w_{\epsilon }(t)+{\int }_0^t{(g_{\epsilon }(\tau )-f_{\epsilon }(\tau )-{\hat{m}}_{\epsilon }(\tau ),v_{\epsilon }(\tau ))}_{2}d\tau ,$$

(17)

where

$$g_{\epsilon }(t)\equiv g(v(t+\epsilon ))-g(v(t)),f_{\epsilon }(t)\equiv f(u(t+\epsilon ))-f(u(t)),$$

$${\hat{m}}_{\epsilon }(t)\equiv m(\parallel \nabla u(t+\epsilon ){\parallel }_2^2)\mathrm{\Delta }u(t+\epsilon )-m(\parallel \nabla u(t){\parallel }_2^2)\mathrm{\Delta }u(t),$$

and

$$m(\parallel \nabla u(t){\parallel }_2^2)\equiv \beta \parallel \nabla u(t){\parallel }_2^{2\gamma }.$$

However, we cannot obtain (16) through the energy equation (17) alone because of the form of the nonlinearity ${\hat{m}}_{\epsilon }(t)$. Hence, we have to work with the auxiliary function:

$$W_{\epsilon }(t)\equiv w_{\epsilon }(t)+\frac{1}{2}m(\parallel \nabla u(t+\epsilon ){\parallel }_2^2)\parallel \nabla u_{\epsilon }(t){\parallel }_2^2.$$

(18)

The corresponding energy equation for W _ε (t) is

$$\begin{array}{ll}{W}_{\epsilon }(0)=W_{\epsilon }(t)& +{\int }_0^t{(g_{\epsilon }(\tau )-f_{\epsilon }(\tau )-m_{\epsilon }(\tau ),v_{\epsilon }(\tau ))}_{2}d\tau \\ -{\int }_0^t{(n_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau ,\end{array}$$

(19)

where

$$m_{\epsilon }(t)\equiv (m(\parallel \nabla u(t+\epsilon ){\parallel }_2^2)-m(\parallel \nabla u(t){\parallel }_2^2))\mathrm{\Delta }u(t)$$

and

$$n_{\epsilon }(t)\equiv m^{\prime }(\parallel \nabla u(t+\epsilon ){\parallel }_2^2){(\mathrm{\Delta }u(t+\epsilon ),v(t+\epsilon ))}_2\mathrm{\Delta }{u}_{\epsilon }(t).$$

Notice that since the solution is uniformly bounded by ${Ê}_0$, there exists a constant ${\tilde{E}}_0>0$, depending on ${Ê}_0$, such that

$$w_{\epsilon }(t)\le W_{\epsilon }(t)\le {\tilde{E}}_0{w}_{\epsilon }(t),$$

(20)

that is, W _ε is an equivalent norm of the solution in H. We shall show the uniform continuity property for W _ε .

For every t ≥ 0, we have either

$$W_{\epsilon }(t+1)\le W_{\epsilon }(t)$$

(21)

or

$$W_{\epsilon }(t+1)>W_{\epsilon }(t).$$

(22)

If (22) holds,

$$\begin{array}{ll}0>W_{\epsilon }(t)& -W_{\epsilon }(t+1)={\int }_t^{t+1}(g_{\epsilon }(\tau )-f_{\epsilon }(\tau )-m_{\epsilon }(\tau ),\\ v_{\epsilon }(\tau ){)}_{2}d\tau -{\int }_t^{t+1}{(n_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau .\end{array}$$

(23)

A well-known inequality can be applied to the monotone form of the damping term:

$${(g_{\epsilon }(t),v_{\epsilon }(t))}_2\ge 2^{2-\lambda }\delta \parallel v_{\epsilon }(t){\parallel }_{\lambda }^{\lambda }.$$

(24)

Therefore, (23) yields

$$\begin{array}{ll}{2}^{2-\lambda }& \delta {\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau <{\int }_t^{t+1}(f_{\epsilon }(\tau )+m_{\epsilon }(\tau ),\\ v_{\epsilon }(\tau ){)}_{2}d\tau +{\int }_t^{t+1}{(n_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau .\end{array}$$

(25)

On the other hand, we apply another inequality for the source term:

$$\left|f_{\epsilon }(t)\right|\le \sigma (r)\mu \left(|u(t+\epsilon ){|}^{r-2}+|u(t){|}^{r-2}\right)|u_{\epsilon }(t)|,$$

(26)

where σ(r) = 1 if r ∈ [2,3] and σ(r) = (r−1)/2 if r > 3.

Now, we apply Hölder inequality to obtain

$$\begin{array}{ll}{\left({\int }_t^{t+1}\parallel f_{\epsilon }(\tau ){\parallel }_2^{2}d\tau \right)}^{\frac{1}{2}}& \le Ĉ\underset{t\ge 0}{sup}a{(u(t))}^{(r-2)/2}\\ \times {\left({\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_{2(r-1)}^{2(r-1)}d\tau \right)}^{\frac{1}{2(r-1)}},\end{array}$$

(27)

where $Ĉ\equiv 2\sigma (r)\mu C(\mathrm{\Omega })$, and C(Ω) > 0 is a Sobolev constant of the injection B ↪ L^2(r−1)(Ω).

We claim that t ↦ u(t) ∈ L^2(r−1)(Ω) must be uniformly continuous. Otherwise, there exists some η₀ > 0 and sequences {ε _n }_n ≥ 1{t _n }_n ≥ 1, such that ε _n  → 0 and t _n  → ∞, as n → ∞, and

$$\parallel u_{{\epsilon }_n}(t_n){\parallel }_{2(r-1)}>{\eta }_0,$$

(28)

for every n ≥ 1. By assumption, B ↪ L^2(r−1)(Ω) is compact, and since {u(t)}_t≥0 is bounded in B, then {u(t _n +ε _n )}_n≥1,{u(t _n )}_n≥1 are precompact in L^2(r−1)(Ω). Therefore, we can extract subsequences ${\{u(t_n^{\prime }+{\epsilon }_n^{\prime })\}}_{n\ge 1},{\{u(t_n^{\prime })\}}_{n\ge 1}$, such that for some fixed n₀, which is sufficiently big, and every n ≥ n₀,

$$\begin{array}{ll}\parallel u(t_n^{\prime }+{\epsilon }_n^{\prime })-u(t_n^{\prime }){\parallel }_{2(r-1)}& \le \parallel u(t_n^{\prime }+{\epsilon }_n^{\prime })-u(t_{n_0}+{\epsilon }_{n_0}){\parallel }_{2(r-1)}\\ +\parallel u(t_{n_0}+{\epsilon }_{n_0})-u{(}_{n_0}){\parallel }_{2(r-1)}\\ +\parallel u(t_{n_0})-u(t_n^{\prime }){\parallel }_{2(r-1)}\\ \le \frac{{\eta }_0}{3}+\frac{{\eta }_0}{3}+\frac{{\eta }_0}{3}={\eta }_0.\end{array}$$

This contradicts (28). Hence, for any η > 0, there exists some $\widehat{\epsilon }(\eta )>0$, such that for every t ≥ 0 and every $\epsilon \in (0,\widehat{\epsilon }(\eta )),$

$${\left({\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_{2(r-1)}^{2(r-1)}d\tau \right)}^{\frac{1}{2(r-1)}}\le {\eta }^{4(\lambda -1)}.$$

(29)

Consequently, from (27), (29), and Hölder inequality, we get

$$\begin{array}{l}{\int }_t^{t+1}\left|{(f_{\epsilon }(\tau ),v_{\epsilon }(\tau ))}_2\right|d\tau \le C{\eta }^{4(\lambda -1)}\\ \times {\left({\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{1}{\lambda }},\end{array}$$

(30)

where C > 0 depends on $Ĉ,{Ê}_0,$ and the inclusion L^λ(Ω) ↪ L²(Ω).

Now notice that

$$\begin{array}{ll}\parallel m_{\epsilon }(t){\parallel }_2& \le \underset{t\ge 0}{sup}\{m^{\prime }(\parallel \nabla u(t){\parallel }_2^2)\}\left|\parallel \nabla u(t+\epsilon ){\parallel }_2^2\right.\\ \left.-\parallel \nabla u(t){\parallel }_2^2\right|\parallel \mathrm{\Delta }u(t){\parallel }_2=\underset{t\ge 0}{sup}\{m^{\prime }(\parallel \nabla u(t){\parallel }_2^2)\}|\\ \left.{(\nabla u(t+\epsilon )+\nabla u(t),\nabla u(t+\epsilon )-\nabla u(t))}_2\right|\parallel \mathrm{\Delta }u(t){\parallel }_2\\ \le \underset{t\ge 0}{sup}\{m^{\prime }(\parallel \nabla u(t){\parallel }_2^2)\parallel \mathrm{\Delta }u(t+\epsilon )+\mathrm{\Delta }u(t){\parallel }_2\parallel \mathrm{\Delta }u(t){\parallel }_2\}\\ \parallel u_{\epsilon }(t){\parallel }_2\le C({Ê}_0)\parallel u_{\epsilon }(t){\parallel }_2,\end{array}$$

then

$$\left|{(m_{\epsilon }(t),v_{\epsilon }(t))}_2\right|\le C({Ê}_0)\parallel u_{\epsilon }(t){\parallel }_2\parallel v_{\epsilon }(t){\parallel }_2.$$

(31)

Also,

$$\begin{array}{ll}\left|{(n_{\epsilon }(t),u_{\epsilon }(t))}_2\right|& \le m^{\prime }(\parallel \nabla u(t+\epsilon ){\parallel }_2^2)\parallel \mathrm{\Delta }u(t+\epsilon ){\parallel }_2\\ \parallel v(t+\epsilon ){\parallel }_2\parallel \mathrm{\Delta }{u}_{\epsilon }(t){\parallel }_2\parallel u_{\epsilon }(t){\parallel }_2\\ \le C({Ê}_0)\parallel u_{\epsilon }(t){\parallel }_2.\end{array}$$

(32)

Here, $C({Ê}_0)>0$ is a constant.

Since B ↪ L²(Ω) is compact, we show like in (29) that

$${\left({\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_2^{2}d\tau \right)}^{\frac{1}{2}}\le {\eta }^{4(\lambda -1)}.$$

(33)

Consequently, from (31) to (33) and Hölder inequality, we obtain

$$\begin{array}{l}{\int }_t^{t+1}\left\{\right|{(m_{\epsilon }(\tau ),v_{\epsilon }(\tau ))}_2|+|{(n_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_2\left|\right\}d\tau \le \hat{C}{\eta }^{4(\lambda -1)}\\ \times \left\{{\left({\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{1}{\lambda }}+1\right\},\end{array}$$

(34)

where $\hat{C}>0$ depends on $C({Ê}_0)$ and the inclusion L^λ(Ω) ↪ L²(Ω).

Taking into account (30) and (34) in (25), we have

$$\begin{array}{l}{\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \le \tilde{C}{\eta }^{4(\lambda -1)}\\ \times \left\{{\left({\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{1}{\lambda }}+1\right\},\end{array}$$

where $\tilde{C}>0$ is a constant. Consequently, for η that is sufficiently small, we obtain

$${\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \le {\eta }^{3\lambda }$$

(35)

and

$${\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_2^{2}d\tau \le \frac{{\eta }^2}{5}.$$

(36)

We want to get a similar estimate for u _ε in the B norm. To this end, from (5), we get

$$\begin{array}{ll}\frac{d}{\mathit{\text{dt}}}{(v_{\epsilon }(t),u_{\epsilon }(t))}_2& -\parallel v_{\epsilon }(t){\parallel }_2^2+\parallel \mathrm{\Delta }{u}_{\epsilon }(t){\parallel }_2^2+\alpha \parallel \nabla u_{\epsilon }(t){\parallel }_2^2\\ +{(g_{\epsilon }(t),u_{\epsilon }(t))}_2\\ ={({\hat{m}}_{\epsilon }(t),u_{\epsilon }(t))}_2+{(f_{\epsilon }(t),u_{\epsilon }(t))}_2.\end{array}$$

(37)

We apply inequality (26) to g _ε , and by Hölder inequality, we get

$$\begin{array}{ll}{\int }_t^{t+1}& {(g_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau \le \sigma (\lambda )\delta {\left({\int }_t^{t+1}\parallel v_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{1}{\lambda }}\\ \times {\left({\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{1}{\lambda }}\left[{\left({\int }_t^{t+1}\parallel v(\tau +\epsilon ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{\lambda -2}{\lambda }}\right.\\ \left.+{\left({\int }_t^{t+1}\parallel v(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{\lambda -2}{\lambda }}\right].\end{array}$$

(38)

Notice that by (6) and (14),

$${\int }_0^{\infty }\parallel v(t){\parallel }_{\lambda }^{\lambda }\mathit{\text{dt}}\le \frac{E_0-E_{\infty }}{\delta }.$$

(39)

By assumption, B ↪ L^λ(Ω), then

$${\left({\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_{\lambda }^{\lambda }d\tau \right)}^{\frac{1}{\lambda }}\le 2C(\mathrm{\Omega })\underset{t\ge 0}{sup}\parallel u(t){\parallel }_B\le C,$$

(40)

where C > 0 depends on the embedding constant C(Ω) and ${Ê}_0$.

Therefore, from (35), (39), and (40) and for η that is sufficiently small, (38) becomes

$$\begin{array}{l}\left|{\int }_t^{t+1}{(g_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau \right|\le 2\sigma (\lambda )\delta C\\ \times {\left(\frac{E_0-E_{\infty }}{\delta }\right)}^{\frac{\lambda -2}{\lambda }}{\eta }^3\le \frac{{\eta }^2}{5}.\end{array}$$

(41)

Now, by (27), (29), and (33) and for small η, we obtain

$${\int }_t^{t+1}{(f_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau \le C{\eta }^{8(\lambda -1)}\le \frac{{\eta }^2}{5}.$$

(42)

Next, we have the estimate

$$\begin{array}{ll}{({\hat{m}}_{\epsilon }(t),u_{\epsilon }(t))}_2& =-m(\parallel \nabla u(t+\epsilon ){\parallel }_2^2)\parallel \nabla u_{\epsilon }(t){\parallel }_2^2\\ +\{m(\parallel \nabla u(t+\epsilon ){\parallel }_2^2)-m(\parallel \nabla u(t){\parallel }_2^2)\}\\ \times {(\mathrm{\Delta }u(t),u_{\epsilon }(t))}_2\\ \le C({Ê}_0)\parallel u_{\epsilon }(t){\parallel }_2.\end{array}$$

Hence, by Hölder inequality, (33) and small η,

$${\int }_t^{t+1}{({\hat{m}}_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau \le \frac{{\eta }^2}{5}.$$

(43)

Therefore, (36), (41), (42), and (43) in (37), yield

$${\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_B^{2}d\tau \le 2\underset{t\ge 0}{sup}\{\parallel v_{\epsilon }(t){\parallel }_2\parallel u_{\epsilon }(t){\parallel }_2\}+\frac{4{\eta }^2}{5}.$$

(44)

Notice that for every t ≥ 0, we have

$$\parallel u_{\epsilon }(t){\parallel }_2\le \epsilon \underset{s\in [t,t+\epsilon ]}{sup}\parallel v(s){\parallel }_2\le \epsilon \sqrt{2{Ê}_0}$$

and

$$\parallel v_{\epsilon }(t){\parallel }_2\le 2\underset{t\ge 0}{sup}\parallel v(t){\parallel }_2\le 2\sqrt{2{Ê}_0}.$$

Consequently, for $\epsilon \in (0,\widehat{\epsilon }(\eta ))$, with $\widehat{\epsilon }(\eta )$ sufficiently small, (44) is

$${\int }_t^{t+1}\parallel u_{\epsilon }(\tau ){\parallel }_B^{2}d\tau \le {\eta }^2.$$

(45)

Hence, in case (22), from (36) and (45) we conclude that

$${\int }_t^{t+1}{w}_{\epsilon }(\tau )d\tau \le \frac{3{\eta }^2}{5}.$$

(46)

Also, from (20) with η small,

$${\int }_t^{t+1}{W}_{\epsilon }(\tau )d\tau \le \frac{3\eta }{5}.$$

(47)

From (19) and by (24), we have for any s ∈ [t,t + 1],t ≥ 0 that

$$\begin{array}{ll}{W}_{\epsilon }(t+1)& \le W_{\epsilon }(s)+{\int }_s^{t+1}{(m_{\epsilon }(\tau )+f_{\epsilon }(\tau ),v_{\epsilon }(\tau ))}_{2}d\tau \\ +{\int }_s^{t+1}{(n_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_{2}d\tau .\\ \le W_{\epsilon }(s)+{\int }_t^{t+1}\left\{\right|{(f_{\epsilon }(\tau ),v_{\epsilon }(\tau ))}_2|+|(m_{\epsilon }(\tau ),\\ v_{\epsilon }(\tau ){)}_2|+|{(n_{\epsilon }(\tau ),u_{\epsilon }(\tau ))}_2\left|\right\}d\tau .\end{array}$$

Then, taking into account (30), (34), and (35) and for η small

$$W_{\epsilon }(t+1)\le W_{\epsilon }(s)+\frac{2\eta }{5}.$$

Therefore, by (47), we obtain

$$W_{\epsilon }(t+1)\le {\int }_t^{t+1}{W}_{\epsilon }(s)\mathit{\text{ds}}+\frac{2\eta }{5}\le \eta .$$

Consequently, in both cases, (21) and (22),

$$W_{\epsilon }(t+1)\le max\{\eta ,W_{\epsilon }(t)\}.$$

Notice that for every t ≥ 0, there exists a natural number N such that N ≤ t ≤ N + 1, then the last estimate implies that

$$W_{\epsilon }(t)\le max\{\eta ,\underset{s\in [N-1,N]}{\text{max}}{W}_{\epsilon }(s)\}.$$

Hence, applied recursively backwards,

$$W_{\epsilon }(t)\le max\{\eta ,\underset{s\in [0,1]}{\text{max}}{W}_{\epsilon }(s)\}.$$

(48)

Since the solution (u,v) : [0,1] → H is uniformly continuous, W _ε has the same property due to (20). Then, (48) implies that t ↦ W _ε (t) is uniformly continuous for any t ≥ 0, and the same holds for the solution in H, again in virtue of (20), that is, we have proven (16).

Precompactness

Next, we shall prove (15), that is, the orbit is a precompact subset of H. We start with {v(t)}_t ≥ 0⊂L²(Ω).

Notice that because of (16),

$$\begin{array}{ll}{∥v(t)-\frac{1}{\epsilon }{\int }_t^{t+\epsilon }v(\tau )d\tau ∥}_2& \le \frac{1}{\epsilon }{\int }_t^{t+\epsilon }\parallel v(t)-v(\tau ){\parallel }_{2}d\tau \\ \le \underset{\tau \in [t,t+\epsilon ]}{\text{sup}}\parallel v(t)-v(\tau ){\parallel }_2\\ \le \sqrt{2\eta }.\end{array}$$

(49)

Since {u(t)}_t ≥ 0 is bounded in B, then

$$\begin{array}{ll}{∥\frac{1}{\epsilon }\left({\int }_t^{t+\epsilon }v(\tau )d\tau \right)∥}_B& \le \frac{1}{\epsilon }\parallel u(t+\epsilon )-u(t){\parallel }_B\\ \le \frac{2}{\epsilon }\underset{t\ge 0}{\text{sup}}\parallel u(t){\parallel }_B\\ \le \frac{2}{\epsilon }\sqrt{2{Ê}_0}.\end{array}$$

Since B ↪ L²(Ω) is compact, ${\{\frac{1}{\epsilon }{\int }_t^{t+\epsilon }v(\tau )d\tau \}}_{t\ge 0}$ is precompact or, equivalently, totally bounded in L²(Ω). Also, by (49), {v(t)}_t≥0 is precompact in L²(Ω).

In a similar way, from (16), we estimate

$${∥u(t)-\frac{1}{\epsilon }{\int }_t^{t+\epsilon }u(\tau )d\tau ∥}_B\le \underset{\tau \in [t,t+\epsilon ]}{\text{sup}}\parallel u(t)-u(\tau ){\parallel }_B\le \sqrt{2\eta }.$$

(50)

We shall prove that ${\{\mathcal{L}{\int }_t^{t+\epsilon }u(\tau )d\tau \}}_{t\ge 0}$ is precompact in B^′ ≡ the dual space of B, where

$$\mathcal{L}\equiv {\mathrm{\Delta }}^2:B\to B^{\prime },$$

then precompactness of {u(t)}_t≥0 in B follows from (50) since

$${\mathcal{L}}^{-1}\equiv {({\mathrm{\Delta }}^2)}^{-1}:B^{\prime }\to B$$

is a linear and continuous operator.

According to the dense and continuous inclusions

$$B\hookrightarrow L^{\lambda }(\mathrm{\Omega })\hookrightarrow L^2(\mathrm{\Omega })\hookrightarrow L^{{\lambda }^{\ast }}(\mathrm{\Omega })\hookrightarrow B^{\prime },$$

where $L^{{\lambda }^{\ast }}(\mathrm{\Omega })={(L^{\lambda }(\mathrm{\Omega }))}^{\prime },{\lambda }^{\ast }=\lambda /(\lambda -1)$, we extend the inner product in L²(Ω) to the duality product in B^′×B. Now, we integrate Equation 1, and since $\mathcal{L}:B\to B^{\prime }$ is closed, we get, in the sense of B^′,

$$\begin{array}{ll}\mathcal{L}{\int }_t^{t+\epsilon }u(\tau )d\tau & =v(t)-v(t+\epsilon )\\ +{\int }_t^{t+\epsilon }M(\parallel \nabla u(\tau ){\parallel }_2^2)\mathrm{\Delta }u(\tau )d\tau \\ -{\int }_t^{t+\epsilon }g(v(\tau ))d\tau +{\int }_t^{t+\epsilon }f(u(\tau ))d\tau .\end{array}$$

(51)

By Hölder inequality and since v(t) is bounded in L²(Ω),

$$\begin{array}{ll}\parallel v(t)-v(t+\epsilon ){\parallel }_{{\lambda }^{\ast }}& \le 2|\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}\underset{t\ge 0}{sup}\parallel (v(t)){\parallel }_2\\ \le 2|\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}C({Ê}_0).\end{array}$$

(52)

By Hölder inequality and (39),

$$\begin{array}{ll}{∥{\int }_t^{t+\epsilon }g(v(\tau ))d\tau ∥}_{{\lambda }^{\ast }}& \le \delta {\int }_t^{t+\epsilon }\parallel (v(\tau )){\parallel }_{\lambda }^{\lambda -1}d\tau \\ \le {\delta }^{\frac{1}{\lambda }}{(E_0-E_{\infty })}^{\frac{\lambda -1}{\lambda }}.\end{array}$$

(53)

By the boundedness of u(t) in B, Hölder inequality and the injection B ↪ L^2(r−1)(Ω), we obtain the estimate

$$\begin{array}{ll}{∥{\int }_t^{t+\epsilon }f(u(\tau ))d\tau ∥}_{{\lambda }^{\ast }}& \le |\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}{∥{\int }_t^{t+\epsilon }f(u(\tau ))d\tau ∥}_2\\ \le \mu |\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}{\int }_t^{t+\epsilon }\parallel (u(\tau )){\parallel }_{2(r-1)}^{r-1}d\tau \\ \le \mu C(\mathrm{\Omega })\underset{t\ge 0}{\text{sup}}\parallel u(t){\parallel }_B,\\ \le \mu C(\mathrm{\Omega })C({Ê}_0).\end{array}$$

(54)

Also,

$$\begin{array}{l}{∥{\int }_t^{t+\epsilon }M(\parallel \nabla u(\tau ){\parallel }_2^2)\mathrm{\Delta }u(\tau )d\tau ∥}_{{\lambda }^{\prime }}\\ \le |\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}{∥{\int }_t^{t+\epsilon }M(\parallel \nabla u(\tau ){\parallel }_2^2)\mathrm{\Delta }u(\tau )d\tau ∥}_2\\ \le \epsilon \left|\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}\underset{t\ge 0}{\text{sup}}\right\{M(\parallel \nabla u(t){\parallel }_2^2)\parallel u(t){\parallel }_B\}\\ \le \epsilon |\mathrm{\Omega }{|}^{\frac{\lambda -2}{2\lambda }}C({Ê}_0),\end{array}$$

(55)

Therefore, (52) to (55) in (51) imply that

$${∥\mathcal{L}{\int }_t^{t+\epsilon }u(\tau )d\tau ∥}_{{\lambda }^{\ast }}\le C,$$

(56)

for some constant C > 0 and every t ≥ 0.

B ↪ L^λ(Ω) is compact by assumption. By Schauder’s theorem (see Brézis [25]), B ↪ L^λ(Ω) is compact if and only if $L^{{\lambda }^{\ast }}(\mathrm{\Omega })\hookrightarrow B^{\prime }$ is compact. Then, (56) implies that ${\{\mathcal{L}{\int }_t^{t+\epsilon }u(\tau )d\tau \}}_{t\ge 0}$ is precompact in B^′. The proof is complete. □

Remark 1

We observe that the main difficulty in the proof of the last theorem is to show precompactness of bounded orbits. This has been accomplished for semilinear wave equations by Haraux [15]. Here, we extend this technique for a Timoshenko equation with nonlinear damping and a source term. This source term has an amplifying effect instead of a restoring one in case of r ≥ 2(γ + 1) (see [23]). Indeed, in another study [23], we proved that, depending on the initial conditions, every solution of (1) either blows up in a finite time or there exists for all time. In this last case, again depending on the initial conditions, the solution is either unbounded or bounded and tends to the set of equilibria $\mathcal{E}$, as time goes to infinity. On the other hand, Theorem 2 shows that every solution of Equation 1 converges to $\mathcal{E}$, whenever 2 < r < 2(γ + 1), that is, when the source term is dominated, then every solution is bounded and tends to the equilibria set as time goes to infinity. Consequently, we give a complete panorama of the qualitative behavior of the solutions of the nonlinear Timoshenko equation, Equation 1. A dynamic analysis of more realistic rod models (see for instance [1]) requires more effort and research.