Exponential growth for wave equation with fractional boundary dissipation and boundary source term
© Dai and Zhang; licensee Springer. 2014
Received: 6 April 2014
Accepted: 24 May 2014
Published: 20 September 2014
The wave equation with boundary source term and fractional boundary dissipation is considered. The exponential growth for sufficiently large initial data is proved. To this end some techniques based on Fourier transforms and some inequalities such as the Hardy-Littlewood-Soblev inequality are used.
Let us mention here that the case in (1)-(4) corresponds to a boundary damping and it has been extensively studied by many authors (see, for instance, – and references therein). It has been proved, in particular, that solutions exist globally in time when the initial data are in a stable set and the solution blows up in finite time when the initial data are in an unstable set.
The problem is motivated by some phenomena in viscoelasticity. Fractional differential systems have proved to be useful in control processing for the last two decades. Recently the linear wave equation with fractionally damped structures has received much attention (see , –) and there has been a growing interest in investigating the solutions and properties of these evolution equation, where the source term vanish. But few contributions are concerned with the nonlinear or semi-linear wave equation with fractionally damped structure. This may be partly attributed to the fact that we do not benefit from a theoretical setting as convenient as the one provided by the semigroup theory . Liang et al. studied the boundary stabilization of a wave equation with fractional order boundary controller by symbolic algebra and numerical similarity. Mbodje  investigated the asymptotic behavior of solutions of the wave equation with a boundary viscoelastic damper of the fractional derivative type by semigroup theory. When the fractional order damped and sources term are the terms of the wave equation, Kirane and Tatar  and Tatar  proved that exponential growth and blow-up result for sufficient large initial data.
The fractional boundary dissipation can also be regarded as a viscosity term and boundary conditions of memory boundary terms. It is worth mentioning here that many authors have considered memory boundary terms (see – and references therein). However, in all these works the kernels appearing in their integral terms are all regular. In our case the kernel is not only singular but also non-integral. In this paper, we prove that the classical energy grows up exponentially when time goes to infinity by means of Fourier transforms and Hardy-Littlewood-Sobolev inequality. This technique has been used successfully by Kirane and Tatar  and Tatar  for the wave equation with fractional order damping. We also pointed out a similar problem with a fractional derivative term on part of its boundary which may blow up in finite time using the different method – and our main idea follows from .
The plan of the paper is as follows: in the next section we will prepare some materials needed to prove our result. Section 2 is devoted to the proof of our main result.
Let, , and, thenwith. Also the mapping fromintois continuous.
3 Exponential growth of the solution
- (1)If , then the Lemma 2.1 with
where depends only m and α.
where and , is the measure of Γ.
where is the Poincaré constant.
- (2)For the case , we note that
- (3)If , we use the estimate
This leads to an estimation of (12). Again the rest of the proof is similar to that in case (1). □
The first author wish to express sincere gratitude to his advisor Dr. Jigen Peng for his constructive and helpful suggestions and his encouragement in the pursuit of this work. This work is supported by National Natural Science Foundation of China (No. 11171311) and by the Natural Science Foundation of Henan Province (1323004100360).
- Bajlekova, EG: Fractional Evolution Equations in Banach Spaces. Dissertation, Eindhoven University of Technology, Eindhoven (2001)MATHGoogle Scholar
- Mbodje B: Wave energy decay under fractional derivative controls. IMA J. Math. Control Inf. 2006, 237: 237-257.MathSciNetMATHGoogle Scholar
- Vitillaro E: Some new results on global nonexistence and blow up for evolution problem with positive initial energy. Rend. Ist. Mat. Univ. Trieste 2000, 31(2):245-275.MathSciNetMATHGoogle Scholar
- Vitillaro E: Global existence for the wave equation with boundary damping and source terms. J. Differ. Equ. 2002, 186: 259-298. 10.1016/S0022-0396(02)00023-2MathSciNetView ArticleMATHGoogle Scholar
- Zhang HW, Hu QY: Asymptotic behavior and nonexistence of wave equation with nonlinear boundary conditions. Commun. Pure Appl. Anal. 2005, 4(4):861-869. 10.3934/cpaa.2005.4.861MathSciNetView ArticleMATHGoogle Scholar
- Liang, J, Chen, YQ, Vinagre, BM, Podlubny, I: Boundary stabilization of a fractional wave equation via a fractional order boundary controller. (2005), [http://www.researchgate.net/publication/228991225_Boundary_stabilization_of_a_fractional_wave_equation_via_a_fractional_order_boundary_controller]Google Scholar
- Matignons D, Audalnet S, Montsney G: Energy decay for wave equations with damping of fractional order. Proc. of the Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena 1998, 638-640. Golden, Colorado, June 1998Google Scholar
- Mbodje B, Montsney G: Boundary fractional derivative control of the wave equation. IEEE Trans. Autom. Control 1995, 40(2):378-382. 10.1109/9.341815View ArticleMATHMathSciNetGoogle Scholar
- Mbodje B, Montsney G, Audounet J: Analysis of fractionally damped flexible systems via a diffusion equation. Int. J. Syst. Sci. 1994, 25(11):1775-1791. 10.1080/00207729408949312View ArticleMATHMathSciNetGoogle Scholar
- Mbodje B, Montsney G, Audounet J, Benchimol P: Optimal control for fractionally damped flexible systems. Proc. of the Third IEEE Conference on Control Applications 1994, 1329-1333. Strathclyde University, Glasgow, 24-26 August 1994 10.1109/CCA.1994.381303Google Scholar
- Kirane M, Tatar N: Exponential growth for fractionally damped wave equation. Z. Anal. Anwend. 2003, 22(1):167-177. 10.4171/ZAA/1137MathSciNetView ArticleMATHGoogle Scholar
- Tatar N: A blow up result for a fractionally damped wave equation. Nonlinear Differ. Equ. Appl. 2005, 12: 215-226. 10.1007/s00030-005-0015-6MathSciNetView ArticleMATHGoogle Scholar
- Kirane M, Tatar N: A memory type boundary stabilization of a mildly damped wave equation. Electron. J. Qual. Theory Differ. Equ. 1999, 6: 1-7.MathSciNetView ArticleMATHGoogle Scholar
- Kirane M, Tatar N: Nonexistence results for a semilinear hyperbolic problem with boundary condition of memory type. Z. Anal. Anwend. 2000, 19(2):1-16. 10.4171/ZAA/961MathSciNetView ArticleMATHGoogle Scholar
- Aassila M: Nonexistence of global solutions of a hyperbolic problem. Math. Comput. Model. 2001, 34: 761-769. 10.1016/S0895-7177(01)00097-8MathSciNetView ArticleMATHGoogle Scholar
- Aassila M, Cavalcanti MM, Soriano JA: Asymptotic stability and energy decay rates for the solutions of the wave equation with memory in a star-shaped domain. SIAM J. Control Optim. 2000, 38: 1587-1602. 10.1137/S0363012998344981MathSciNetView ArticleMATHGoogle Scholar
- Labidi S, Tatar N: Unboundedness for the Euler-Bernoulli beam equation with a fractional boundary dissipation. Appl. Math. Comput. 2005, 161: 697-706. 10.1016/j.amc.2003.12.057MathSciNetView ArticleMATHGoogle Scholar
- Labidi S, Tatar N: Blow up for the Euler-Bernoulli beam problem with a fractional boundary dissipation. Dyn. Syst. Appl. 2008, 17(1):109-119.MathSciNetMATHGoogle Scholar
- Labidi S, Tatar N: Blow-up of solutions for a nonlinear beam equation with fractional feedback. Nonlinear Anal. 2011, 74: 1402-1409. 10.1016/j.na.2010.10.012MathSciNetView ArticleMATHGoogle Scholar
- Lu LQ, Li SJ: Blow up of positive initial energy solutions for a wave equation with fractional boundary dissipation. Appl. Math. Lett. 2011, 24: 1729-1734. 10.1016/j.aml.2011.04.030MathSciNetView ArticleMATHGoogle Scholar
- Hormander L: The Analysis of Linear Partial Differential Operators. I. Springer, Berlin; 1983.MATHGoogle Scholar
- Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton; 1993.MATHGoogle Scholar
- Lions JL, Magenes E: Non-Homogeneous Boundary Value Problem and Applications. Springer, Berlin; 1983.MATHGoogle Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
- Greenberg G, Londen SO, Staffans O: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge; 1990.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.