Journal of Inequalities and Applications

Impact Factor 0.791

Journal of Inequalities and Applications Cover Image
Research
Open Access

Asymptotic behavior of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms

Journal of Inequalities and Applications20162016:318

https://doi.org/10.1186/s13660-016-1269-8

©  The Author(s) 2016

Received: 11 March 2016

Accepted: 26 September 2016

Published: 6 December 2016

Abstract

We study the long time asymptotic behavior of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms. By using the integral estimation method combined with the Gronwall inequality, we point out that the global strong solutions of the problems decay to zero exponentially with the passage of time to infinity. The proof is rigorous and only based on some relatively weak assumptions on the nonlinear term.

Keywords

nonlinear evolution equation dispersive dissipative asymptotic behavior

MSC

35L35 35L75 35B40

1 Introduction

Nonlinear pseudo-hyperbolic equations were proposed from biological and mechanical problems in recent years, such as nerve conduction and the longitudinal vibration of rods with viscous effects, which have important practical and theoretical backgrounds.

More and more attention is paid to nonlinear evolution equations with dispersive and dissipative terms because these problems arise widely in practical applications. In 1980, Zhu [1] investigated the communication of longitudinal deformation wave for a flexible rod, he considered the influence of fourth-order nonlinear evolution equations with nonlinear, dispersive and dissipative terms,
$$u_{tt}-C_{0}^{2} \bigl(1+na_{n}u_{x}^{n-1} \bigr)u_{xx}-\beta u_{xxtt}=\gamma u_{xxt}, $$
where \(C_{0}, \gamma>0, \beta>0\), and \(a_{n}\neq0\) are constants. The \(C_{0}\) represents the phase velocity of zero frequency linear wave. In [1–3] Zhu, Zhang, and Yang discussed the solitary wave solutions of the equations. In [4, 5] Saxton and Hrusa discussed the well-posedness of local solution and the existence of a global solution. In [6] Liu and Zhao considered the global existence of \(W^{k,p}\) solutions to wave equations with a dispersive term. The main part of the equation for their research is \(u_{tt}-\alpha u_{xxt}-u_{xxtt}=\sigma(u_{x})_{x}\), the initial conditions were mentioned in [6].

In the research of elasto-plastic-microstructure models we mention the equation with a nonlinear term \(\sigma({u^{2}_{x}})_{x} +f(u)\) [7]. In [8] the authors discussed the initial boundary value problem with nonlinear term \(\sigma({u_{x}})_{x} +f(u)\). They obtained the global existence and blow-up of solutions. Therefore, these nonlinear evolution equations with different nonlinear terms such as \(\sigma({u_{x}})_{x} +f(u)\) have a strong physical background. The existence of a global solution has many classical results to the problem [9–13].

It is well known that the stability of differential equation is associated with problems that are described by the development of long time behavior. When the global properties of the solutions have been well studied, researchers in various fields naturally place stability as the focus. The asymptotic behavior of solutions is an important part of the stability content, there is some work on these equations. In [14], Shang investigated the initial and boundary value problem; by using the Galerkin method and the energy estimate method, the author not only obtained the whole existence and uniqueness of strong solution but also considered the asymptotic behavior and blow-up of solutions. In [15], the authors considered the asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order. For other relevant work, we refer to [16–19].

This study, based on the previous research, analyzes the global asymptotic behavior of strong solutions to problem (1)-(3):
$$\begin{aligned}& u_{tt}-u_{xx}-\alpha{u_{xxt}}-u_{xxtt}= \sigma({u_{x}})_{x} +f(u),\quad 0< x< 1, t>0, \end{aligned}$$
(1)
$$\begin{aligned}& u(x,0)=u_{0}(x), \qquad {u_{t}(x,0)=u_{1}(x)},\quad 0< x< 1, \end{aligned}$$
(2)
$$\begin{aligned}& u(0,t)=u(1,t)=0,\quad t \geq0, \end{aligned}$$
(3)
by using the integral estimation method and the Gronwall inequality. The result is the further generalization of some existing results. Here \(\alpha\geq0\) is a constant, \(\sigma(u)\) is a rational function of u, f is a given nonlinear smooth function.

Throughout this paper, let \(\|\cdot\|_{p}\) stand for the norm \(\|\cdot\|_{L^{p}(0,1)}\), \(\|\cdot\|\) stand for \(\|\cdot\|_{2}\), and let the inner product be \((u,v)=\int_{0}^{1}uv \,\mathrm{d}x\).

2 Preliminaries

Lemma 2.1

(Gronwall inequality) [20]

Assume that \(y(t)\in L^{1}(0,T)\) and there exist positive constants a and b, such that
$$y(t)\leq a+b \int_{0}^{T}y(z)\,\mathrm{d}z,\quad 0\leq t\leq T, $$
then we have
$$y(t)\leq ae^{bt},\quad 0\leq t\leq T. $$
In order to obtain the asymptotic behavior of solutions with dispersive and dissipative terms to problem (1)-(3), we first introduce the energy as follows:
$$E(t)=\frac{1}{2} \bigl(\|u_{t}\|^{2}+ \|u_{x} \|^{2}+\|u_{xt}\|^{2} \bigr)+ \int_{0}^{1} \bigl( \varPhi(u_{x})-F(u) \bigr)\,\mathrm{d} x, $$
where
$$\varPhi(s)= \int_{0}^{s}\sigma(\tau)\,\mathrm{d}\tau,\qquad F(u)= \int_{0}^{u}f(s)\,\mathrm{d}s. $$

3 Main results

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.

Declarations

Acknowledgements

The authors would link to express their deep gratitude to the referee for his/her carefully reading and invaluable comments. This work was supported by the Heilongjiang Province Education Office Science and Technology Research Foundation (No. 12541869), Natural Science Foundation of Heilongjiang Province (No. A201420) and Qiqihar University Young League Science Foundation (Nos. 2014K-M15 and 2014K-Z03).

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, Qiqihar University
(2)
Department of Mathematics, Harbin Institute of Technology
(3)
Department of Mathematics, Heilongjiang University of Science and Technology

References

  1. Zhu, WQ: Nonlinear waves in elastic rods. Acta Mech. Solida Sin. 16(2), 247-253 (1980) Google Scholar
  2. Zhuang, W, Yang, GT: The propagation of solitary waves in a nonlinear elastic rod. Appl. Math. Mech. 7(7), 615-626 (1986) View ArticleMATHGoogle Scholar
  3. Zhang, SY, Zhuang, W: Strain solitary waves in the nonlinear elastic rods. Acta Mech. Sin. 20(1), 58-66 (1988) Google Scholar
  4. Saxton, R: Existence of solutions for a finite nonlinearly hyperelastic rod. J. Math. Anal. Appl. 105(1), 59-75 (1985) MathSciNetView ArticleMATHGoogle Scholar
  5. Hrusa, WJ: Global existence and asymptotic stability for a semi-linear hyperbolic Volterra equation with large initial data. SIAM J. Math. Anal. 16(1), 110-134 (1985) MathSciNetView ArticleMATHGoogle Scholar
  6. Liu, YC, Zhao, JS: Global existence of \(W^{k,p}\) solutions for a class of nonlinear wave equations with dispersive term. Nonlinear Anal. 62, 45-63 (2005) MathSciNetView ArticleGoogle Scholar
  7. An, L, Peirce, A: A weakly nonlinear analysis of elasto-plastic-microstructure models. SIAM J. Appl. Math. 55(1), 136-155 (1995) MathSciNetView ArticleMATHGoogle Scholar
  8. Chen, GW, Yang, ZJ: Existence and non-existence of global solutions for a class of non-linear wave equations. Math. Methods Appl. Sci. 23(7), 615-631 (2000) MathSciNetView ArticleMATHGoogle Scholar
  9. Yang, ZJ, Song, CM: On the problem of the existence of global solutions for a class of nonlinear evolution equations. Acta Math. Appl. Sin. 20(3), 321-331 (1997) MathSciNetMATHGoogle Scholar
  10. Liu, YC, Liu, DC: Initial and boundary value problem of equation \(u_{tt}=u_{xxt}+\sigma (u_{x} )_{x}\). J. Huazhong Univ. Sci. Technol. 16(6), 169-173 (1988) Google Scholar
  11. Chen, G, Rui, W, Chen, X: Cauchy problem for a damped generalized IMBq equation. J. Math. Phys. 52(5), 053504 (2011) MathSciNetView ArticleMATHGoogle Scholar
  12. Geng, SF, Chen, G: Global existence of solutions of Cauchy problem for generalized Bbm-Burgers equation. Acta Math. Sci. 33(4), 1007-1023 (2013) MathSciNetView ArticleMATHGoogle Scholar
  13. Chen, G, Song, R, Wang, S: Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation. J. Math. Anal. Appl. 368(1), 19-31 (2010) MathSciNetView ArticleMATHGoogle Scholar
  14. Shang, YD: Initial boundary value problem for equations \(u_{tt}-\vartriangle u- \vartriangle u_{t}-\vartriangle u_{tt}=f(u)\). Acta Math. Appl. Sin. 23(23), 385-393 (2000) MATHGoogle Scholar
  15. Chen, X, Chen, G: Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order. Nonlinear Anal., Theory Methods Appl. 68(4), 892-904 (2008) MathSciNetView ArticleMATHGoogle Scholar
  16. Linares, F, Ponce, G: Asymptotic behavior of solutions for the k-gKdV equations. In: Introduction to Nonlinear Dispersive Equations, pp. 191-214 (2015) Google Scholar
  17. Kawashima, S, Wang, YZ: Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation. Anal. Appl. 13(3), 233-254 (2015) MathSciNetView ArticleMATHGoogle Scholar
  18. Xu, RZ, Liu, YC: Asymptotic behavior of solutions for initial-boundary value problems for strongly damped nonlinear wave equations. Nonlinear Anal. 69, 2492-2495 (2008) MathSciNetView ArticleMATHGoogle Scholar
  19. Xu, RZ, Zhao, XR, Shen, JH: Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms. Appl. Math. Mech. 29(2), 259-262 (2008) MathSciNetView ArticleMATHGoogle Scholar
  20. Adams, RA, Fournier, JJ: Sobolev Spaces. Academic Press, New York (2003) MATHGoogle Scholar

Copyright

© The Author(s) 2016