Theorem 3.1
Assume that
\(0\leq\varPhi(s)\leq k_{1}\sigma(s)s\), \(0\leq-F(u)\leq-k_{2}f(u)u\), \(\forall s \in\mathbb{R}\), where
\(k_{i},i=1,2\), are two nonnegative constants. Let
\(u(x,t)\)
be the global strong solution to problem (1)-(3), for positive constants
λ
and
C, satisfying the inequality
$$ \|u_{t}\|^{2}+\|u_{x} \|^{2}+\|u_{xt}\|^{2} +2 \int_{0}^{1} \bigl( \varPhi (u_{x})-F(u) \bigr) \,\mathrm{d}x\leq{CE}(0)e^{-\lambda t},\quad 0\leq t< \infty. $$
(4)
Proof
We multiply the PDE (1) by \(u_{t}\) and integrate over \((0,1)\),
$$\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \frac{1}{2} \bigl(\|u_{t} \|^{2}+\|u_{x}\|^{2}+\| u_{xt} \|^{2} \bigr)+ \int_{0}^{1} \bigl( \varPhi(u_{x})-F(u) \bigr) \,\mathrm{d}x \biggr) +\alpha\|u_{xt}\|^{2}=0, $$
that is,
$$ \frac{\mathrm{d}}{\mathrm{d}t}E(t)+\alpha\| u_{xt} \|^{2}=0. $$
(5)
Take \(\delta>0\), multiply equality (5) by \(e^{\delta t}\) to obtain
$$ \frac{\mathrm{d}}{\mathrm{d}t} \bigl(e^{\delta t}E(t) \bigr)+\alpha e^{\delta t}\|u_{xt}\|^{2}=\delta e^{\delta t}E(t). $$
(6)
We integrate (6) on t from 0 to t, then we obtain the equality
$$\begin{aligned} &e^{\delta t} E(t)+\alpha \int_{0}^{t}e^{\delta\tau}\|u_{x\tau} \|^{2}\,\mathrm{d}\tau \\ &\quad= E(0)+\delta \int_{0}^{t}e^{\delta\tau}E(\tau)\,\mathrm{d}\tau \\ &\quad= E(0)+\frac{\delta}{2} \int_{0}^{t} e^{\delta\tau} \bigl( \|u_{\tau}\|^{2}+\| u_{x\tau}\|^{2} \bigr)\, \mathrm{d}\tau \\ &\qquad{}+\delta \int_{0}^{t}e^{\delta\tau} \biggl( \frac{1}{2}\|u_{x}\|^{2}+ \int_{0}^{1} \bigl( \varPhi(u_{x})-F(u) \bigr) \,\mathrm{d}x \biggr)\,\mathrm{d}\tau. \end{aligned}$$
(7)
Since \(-F(u)\geq0\), we obtain \(E(t)\geq0\) as \(0\leq t<\infty\). Because \(0\leq\varPhi(s)\leq k_{1}\sigma(s)s\), \(0\leq-F(u)\leq-k_{2}f(u)u\) combined with equality (1) we get
$$\begin{aligned} &\frac{1}{2}\|u_{x}\|^{2}+ \int_{0}^{1} \bigl( \varPhi(u_{x})-F(u) \bigr) \,\mathrm{d}x \\ &\quad\leq k \bigl(\|u_{x}\|^{2}-(u_{tt}-u_{xx}-u_{xxtt}- \alpha u_{xxt},u) \bigr) \\ &\quad=k \bigl( \|u_{x}\|^{2}-(u_{tt},u)+(u_{xx},u)+(u_{xxtt},u)+ \alpha (u_{xxt},u) \bigr) \\ &\quad=-k \biggl((u_{tt},u)+(u_{xtt},u_{x})+ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\| u_{x}\|^{2} \biggr), \end{aligned}$$
(8)
where \(k=\max\{1/2,k_{1},k_{2}\}\). Hence we have
$$ \begin{aligned}[b] &\int_{0}^{t} e^{\delta\tau} \biggl( \frac{1}{2}\| u_{x}\|^{2}+ \int_{0}^{1} \bigl( \varPhi(u)-F(u) \bigr) \, \mathrm{d}x \biggr) \,\mathrm{d}\tau\\ &\quad\leq -k \int_{0}^{t}e^{\delta\tau} \biggl( (u_{\tau\tau},u)+(u_{x\tau\tau },u_{x})+\frac{\alpha}{2} \frac{\mathrm{d}}{\mathrm{d}\tau}\| u_{x}\|^{2} \biggr) \,\mathrm{d}\tau. \end{aligned} $$
(9)
By integration by parts, we get
$$\begin{aligned} &{-} \int_{0}^{t} e^{\delta\tau}(u_{\tau\tau},u)\, \mathrm{d}\tau \\ &\quad=-e^{\delta t}(u_{t},u)+(u_{1},u_{0}) +\delta \int_{0}^{t}e^{\delta\tau}(u_{\tau},u)\, \mathrm{d}\tau+ \int_{0}^{t}e^{\delta \tau}\|u_{\tau}\|^{2}\,\mathrm{d}\tau \\ &\quad\leq\frac{1}{2}e^{\delta t} \bigl(\|u_{t}\|^{2}+ \|u\|^{2} \bigr)+\frac{1}{2} \bigl(\|u_{1} \|^{2}+\|u_{0}\| ^{2} \bigr) \\ &\qquad{}+\frac{1}{2}\delta \int_{0}^{t}e^{\delta\tau}(\|u_{\tau}) \|^{2}+\|u\|^{2}) \,\mathrm{d}\tau+ \int_{0}^{t}e^{\delta\tau}\|u_{\tau}\|^{2}\,\mathrm{d}\tau, \end{aligned}$$
(10)
$$\begin{aligned} &{-} \int_{0}^{t} e^{\delta\tau}(u_{x\tau\tau},u_{x}) \,\mathrm{d}\tau \\ &\quad=-e^{\delta t}(u_{xt},u_{x})+(u_{x1},u_{x0}) +\delta \int_{0}^{t}e^{\delta\tau}(u_{x\tau},u_{x}) \,\mathrm{d}\tau+ \int _{0}^{t}e^{\delta\tau}\|u_{x\tau} \|^{2}\,\mathrm{d}\tau \\ &\quad\leq\frac{1}{2}e^{\delta t} \bigl(\|u_{xt} \|^{2}+\|u_{x}\|^{2} \bigr)+\frac{1}{2} \bigl( \|u_{x1}\| ^{2}+\|u_{x0}\|^{2} \bigr) \\ &\qquad{}+\frac{\delta}{2} \int_{0}^{t}e^{\delta\tau} \bigl( \|u_{x\tau}\|^{2}+\|u_{x}\|^{2} \bigr)\, \mathrm{d}\tau+ \int_{0}^{t}e^{\delta\tau} \|u_{x\tau} \|^{2}\,\mathrm{d}\tau, \end{aligned}$$
(11)
where \(u_{x1}=u_{x}(x,1), u_{x0}=u_{x}(x,0)\), and
$$\begin{aligned} -\frac{1}{2} \int_{0}^{t} e^{\delta\tau}\frac{\mathrm{d}}{\mathrm{d}\tau} \|u_{x}\| ^{2}\,\mathrm{d}\tau &=-\frac{1}{2}e^{\delta t}\|u_{x}\|^{2}+ \frac{1}{2}\|u_{x0}\|^{2}+\frac{\delta}{2} \int _{0}^{t}e^{\delta\tau}\|u_{x} \|^{2}\,\mathrm{d}\tau \\ &\leq\frac{1}{2}\|u_{x0}\|^{2}+\frac{\delta}{2} \int_{0}^{t}e^{\delta\tau}\|u_{x}\| ^{2}\,\mathrm{d}\tau. \end{aligned}$$
(12)
Inserting (10)-(12) into (9), we have
$$\begin{aligned} &{-}k \int_{0}^{t} e^{\delta\tau} \biggl( (u_{\tau\tau},u)+(u_{x\tau\tau },u_{x})+\frac{\alpha}{2} \frac{\mathrm{d}}{\mathrm{d}\tau}\| u_{x}\|^{2} \biggr) \,\mathrm{d}\tau \\ &\quad= -k e^{\delta t} \biggl( (u_{t},u)+(u_{xt},u_{x})+ \frac{\alpha}{2}\frac {\mathrm{d}}{\mathrm{d}t}\| u_{x}\|^{2} \biggr) \\ &\qquad{} -k \biggl( (u_{1},u_{0})+(u_{x1},u_{x0})+ \frac{\alpha}{2}\| u_{x0}\|^{2} \biggr) -k \int_{0}^{t}e^{\delta\tau} \bigl( \| u_{\tau}\|^{2}+\| u_{x\tau}\|^{2} \bigr) \, \mathrm{d}\tau \\ &\qquad{}-\delta k \int_{0}^{t}e^{\delta\tau} \biggl( (u_{\tau},u)+(u_{x\tau },u_{x})+\frac{\alpha}{2} \frac{\mathrm{d}}{\mathrm{d}\tau}\| u_{x}\|^{2} \biggr) \,\mathrm{d}\tau \\ &\quad\leq M, \end{aligned}$$
(13)
where
$$\begin{aligned}[b] M={}&k \int_{0}^{t}e^{\delta\tau} \bigl( \| u_{\tau}\|^{2}+\| u_{x\tau}\| ^{2} \bigr) \, \mathrm{d}\tau \\ &{}+ \frac{k}{2}e^{\delta t} \bigl( \| u_{t} \|^{2}+\| u\|^{2}+(1+\alpha)\| u_{x} \|^{2}+\| u_{xt}\|^{2} \bigr) \\ &{}+ \frac{k}{2}e^{\delta t} \bigl( \| u_{1} \|^{2}+\| u_{0}\|^{2}+(1+\alpha)\| u_{x0} \|^{2}+\| u_{x1}\|^{2} \bigr) \\ &{} + \frac{k}{2}\delta \int_{0}^{t}e^{\delta\tau} \bigl( \| u_{\tau}\|^{2}+\| u\| ^{2}+(1+\alpha)\| u_{x}\|^{2}+\| u_{x\tau}\|^{2} \bigr) \, \mathrm{d}\tau. \end{aligned} $$
By substituting of the inequality (13) into (7) and using the Poincaré inequality, there exist positive constants \(C_{0}\) and \(C_{1}\), and we find
$$\begin{aligned} &\mathrm{e}^{\delta t} E(t)+\alpha \int_{0}^{t}\mathrm{e}^{\delta\tau} \|u_{x\tau }\|^{2}\,\mathrm{d}\tau \\ &\quad \leq C_{0}E(0)+\frac{\delta}{2} \int_{0}^{t}\mathrm{e}^{\delta\tau} \bigl( \|u_{\tau}\| ^{2}+\|u_{x\tau}\|^{2} \bigr) \, \mathrm{d}\tau+C_{1}\delta\mathrm{e}^{\delta t} E(t) +C_{1}\delta^{2} \int_{0}^{t}\mathrm{e}^{\delta\tau}E(\tau)\, \mathrm{d}\tau \\ &\quad\leq C_{0}E(0)+\frac{\delta}{2}(1+\lambda_{1}) \int_{0}^{t}\mathrm{e}^{\delta\tau} \|u_{x\tau}\|^{2}\,\mathrm{d}\tau+C_{1}\delta \mathrm{e}^{\delta t}E(t)+C_{1}\delta ^{2} \int_{0}^{t}\mathrm{e}^{\delta\tau}E(t)\, \mathrm{d}\tau, \end{aligned}$$
(14)
where
$$\lambda_{1}= \mathop{\sup_{u\in H_{0}^{1}(\Omega)}}_{u\neq0} \frac{\|u\|^{2}}{\|u_{x}\|^{2}}. $$
Take δ, such that
$$0< \delta< \min \biggl\{ \frac{2\alpha}{1+\lambda_{1}},\frac{1}{2C_{1}} \biggr\} . $$
Then combining with (14) we can obtain
$$\begin{aligned} \mathrm{e}^{\delta T}E(t)\leq2 C_{0}E(0)+2C_{1} \delta^{2} \int_{0}^{t}\mathrm{e}^{\delta\tau}E(\tau)\, \mathrm{d}\tau, \end{aligned}$$
(15)
which according to the Gronwall inequality represented by Lemma 2.1 leads to
$$\begin{aligned} \mathrm{e}^{\delta t}E(t)\leq2C_{0}E(0)\mathrm{e}^{2C_{1}\delta^{2}t}, \quad 0\leq t< \infty, \end{aligned}$$
(16)
and
$$\begin{aligned} E(t)\leq2C_{0}E(0)\mathrm{e}^{-\lambda t}, \quad 0\leq t< \infty, \end{aligned}$$
(17)
that is,
$$\|u_{t}\|^{2}+\|u_{x}\|^{2}+ \|u_{xt}\|^{2} +2 \int_{0}^{1} \bigl(\varPhi (u_{x})-F(u) \bigr) \,\mathrm{d}x\leq{CE}(0)e^{-\lambda t},\quad 0\leq t< \infty, $$
where \(\lambda=\delta(1-2C_{1}\delta)>0, C=2C_{0}\).
The proof is complete. □
Corollary 3.1
If the conditions of Theorem
3.1
are satisfied, the following result for the global strong solution
u
of problem (1)-(3):
$$\|u_{t}\|^{2}+\|u_{x}\|^{2}+ \|u_{xt}\|^{2} +2 \int_{0}^{1}\varPhi(u_{x})\,\mathrm{d}x \leq 2{CE}(0)e^{-\lambda t},\quad 0\leq t< \infty, $$
still holds.
Corollary 3.2
If (1) is replaced by the following:
$$u_{tt}-u_{xx}-\alpha{u_{xxt}}-\beta u_{xxtt}=\sigma({u_{x}})_{x}+f(u), $$
then the conclusion of Theorem
3.1
still holds. Here
α
and
β
are positive constants.
This paper has investigated the asymptotic behavior of the global strong solution to a class of fourth-order nonlinear evolution equations with both dispersive and dissipative terms. By using the multiplier method and the integral estimate methods, we prove that the global strong solutions of the problem decay to zero exponentially as the time tends to infinity, under weaker conditions regarding the nonlinear term. It should be pointed out that the method in the present paper can also be extended to the case when the initial-boundary value problem of nonlinear evolution equations are multidimensional. For this case, a different expression for the asymptotic behavior will be employed. This work will be left for our future research.