Skip to main content

Exponential growth for wave equation with fractional boundary dissipation and boundary source term

Abstract

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.

1 Introduction

In this paper we consider the unboundedness of the classical energy for the following problem:

u t t =Δu,xΩ,t>0,
(1)
u=0,x Γ 0 ,t>0,
(2)
u v + t α u= | u | m 1 u,x Γ 1 ,t>0,
(3)
u(x,0)= u 0 (x), u t (x,0)= u 1 (x),xΩ,
(4)

where Ω is a bounded domain in R n with smooth boundary Γ such that Γ= Γ ¯ 0 Γ ¯ 1 =ϕ, m>1 is a constant. The initial data u 0 (x) and u 1 (x) are given function, v denotes the outward normal derivative. The notation t α stands for the Caputo fractional derivative of order α with respect to the time variable [1], [2]. It is defined as follows:

t α ω(t)= 1 Γ ( 1 α ) 0 t ( t s ) α ω t (s)ds,0<α<1.

Let us mention here that the case α=0 in (1)-(4) corresponds to a boundary damping and it has been extensively studied by many authors (see, for instance, [3]–[5] 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 [2], [6]–[10]) 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 [10]. Liang et al.[6] studied the boundary stabilization of a wave equation with fractional order boundary controller by symbolic algebra and numerical similarity. Mbodje [2] 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 [11] and Tatar [12] 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 [13]–[16] 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 [11] and Tatar [12] 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 [17]–[20] and our main idea follows from [12].

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.

2 Preliminaries

Throughout this paper, we denote by H m (Ω), L p (Ω) the usual Sobolev space, and

(u,v)(t)= Ω u(x,t)v(x,t)dt, u 2 =(u,u)(t), u p , Γ p = Γ | u ( x ) | p dΓ.

Let us define

E(t)=E ( u ( t ) ) = 1 2 u t 2 + 1 2 u 2 1 m + 1 u m + 1 , Γ m + 1 .
(5)

It is easily seen that

d E ( t ) d t = 1 Γ ( 1 α ) Γ u t 0 t ( t s ) α u t (s)dsdΓ.
(6)

Observe that d E ( t ) d t is of an undefined sign and then the decreasing of the energy is not guaranteed. However, an integration of (6) with respect to time yields

E(t)E(0)= 1 Γ ( 1 α ) 0 t Γ u t (s) 0 s ( s z ) α u t (z)dzdxds0.
(7)

Hence E(t) is uniformly bounded by E(0), this is

E(t)E(0),t0.

Lemma 2.1

(Hardy-Littlewood-Sobolev inequality, [21], p.354 in [22])

Letu L p (R), p>1, 0<λ<1andλ>1 1 p , then( 1 | x | λ )u L q (R)with 1 q =λ+ 1 p 1. Also the mapping fromu L p (R)into( 1 | x | λ )u L q (R)is continuous.

Lemma 2.2

([23])

In the subspace

V= { v H 1 ( Ω ) , v = 0  on  Γ 0 }

of the Sobolev space H 1 (Ω), there exists a constant C 0 >0such that

v 2 C 0 v 2 ,vV.

Lemma 2.3

([1], [24])

If we denote t β Γ ( 1 β ) by k β (t), then we have

k r + η 1 (t)=( k r k η )(t),0<r,η<1.

3 Exponential growth of the solution

Theorem 3.1

Letu(x,t)be a regular solution of (1)-(4). If the initial dataE(0)is large enough, m>1, then the solutionu(x,t)grows up exponentially in the L m + 1 (Γ)-norm.

Proof

Let us define

G(t)=E(t)δ(u, u t ),

where 0<δ<1 is a small constant to be determined later. Multiplying (1) by u t and then integrating over Ω, we get

d d t ( E ( t ) δ ( u , u t ) ) + 1 Γ ( 1 α ) Γ u t ( t ) 0 t ( t s ) α u t ( s ) d s d Γ = δ Γ ( 1 α ) Γ u 0 t ( t s ) α u t ( s ) d s d Γ δ u t 2 δ u m + 1 , Γ m + 1 + δ u 2 .
(8)

For simplicity, we denote

I 1 = 1 Γ ( 1 α ) 0 T u ( s ) 0 s ( s z ) α u t ( z ) d s d z , I 2 = 1 Γ ( 1 α ) 0 T u t ( s ) 0 s ( s z ) α u t ( z ) d s d z .

Then, using the definition of G(t) and integrating (8) over [0,t] for t, we may write

G(t)G(0)+ I 2 =δ I 1 δ 0 t u t 2 dsδ 0 t u m + 1 , Γ m + 1 ds+δ 0 t u 2 ds.
(9)

Next, we estimate, for a fixed t=T>0,

L T ω(τ)= { ω ( τ ) , τ [ 0 , T ] , 0 , τ R [ 0 , T ] ,

and

L κ β (τ):= { κ β ( τ ) , τ > 0 , 0 , τ 0 ,

where κ β (τ) is as in Lemma 2.3. Then we can easily see that

I 1 = + L T u ( s ) + L κ α ( s z ) L T u t ( z ) d z d s = + F ( L T u ) ( σ ) F ( L κ α L T u t ) ¯ ( σ ) d σ ,
(10)

where we have used the Parseval theorem [21] and denoted by F(f) the usual Fourier transform of f. Then, using Lemma 2.3, the Cauchy-Schwarz inequality, and the Young inequality, we obtain for ρ 1 >0

I 1 ( + | F ( L κ α + 1 2 ) F ( L T u ) | 2 d σ ) 1 2 ( + | F ( L κ α + 1 2 ) F ( L T u t ) | 2 d σ ) 1 2 ρ 1 + | F ( L κ α + 1 2 ) F ( L T u t ) | 2 d σ + 1 4 ρ 1 + | F ( L κ α + 1 2 ) F ( L T u ) | 2 d σ .
(11)

For the last term in (11), we have

I = + | F ( L κ α + 1 2 ) F ( L T u ) | 2 d σ 1 cos ( α 1 2 π ) + L T u ( s ) ( L κ α L T u ) ( s ) d s 1 cos ( α 1 2 π ) ( + | L T u | m + 1 d s ) 1 m + 1 ( + ( L κ α L T u ) m + 1 m ( s ) d s ) m m + 1 ;
(12)

here we have used the Hölder inequality and the following inequality [25], Theorem 16.5.1]:

+ | F ( κ β ) F ( u ) | 2 dσ 1 cos ( ( β 1 ) π ) + u(s)( k 2 β 1 u)(s)ds.

Next we consider three cases.

  1. (1)

    If m m + 1 <α<1, then the Lemma 2.1 with

    q= m m + 1 ,λ=α,p=r= m + 1 2 m + 1 α ( m + 1 ) >1

implies that

( + ( L κ ( α + 1 ) L T u ) m + 1 m ( s ) d s ) m m + 1 C 1 Γ ( 1 α ) ( + | L T u | r d s ) 1 r ,
(13)

where C 1 = C 1 (m,α) depends only m and α.

Since r<2<m+1, it is easily to see that

( + | L T u | r d s ) 1 r T m + 1 r m + 1 ( 0 T | u ( s ) | m + 1 d s ) 1 m + 1 = T m + 1 r m + 1 ( + | L T u | m + 1 ( s ) d s ) 1 m + 1 .
(14)

Thus, from (12)-(14), we deduce that

I C 1 Γ ( 1 α ) cos ( α 1 2 π ) T m + 1 r m + 1 ( + | L T u | m + 1 ( s ) d s ) 1 m + 1 2 m + 1 C 1 m + 1 2 [ Γ ( 1 α ) cos ( α 1 2 π ) ] m + 1 2 + | L T u | m + 1 d s + m 1 m + 1 T m + 1 r m + 1 .
(15)

Using the estimate (15) in (11), we have

I 1 ρ 1 + | F ( L κ α + 1 2 ( s z ) ) F ( L T u t ) | 2 d σ + M ρ 1 + | L T u | m + 1 d s + m 1 4 ρ 1 ( m + 1 ) T m + 1 r m 1 = ρ 1 0 T | κ α + 1 2 u t | 2 d s + M ρ 1 0 T u m + 1 d s + m 1 4 ρ 1 ( m + 1 ) T m + 1 r m 1 ,
(16)

where M= 1 2 ( m + 1 ) [ Γ ( 1 α ) C 1 cos ( α + 1 2 π ) ] m + 1 2 . Analogously,

I 2 + L T u t ( s ) + L κ α ( s z ) ( L T u t ) ( z ) d z d s = + F ( L T u t ) ( σ ) F ( L κ α L T u t ) ¯ ( σ ) d σ + | F ( L κ α + 1 2 ) F ( L T u t ) | 2 d σ = 0 T | κ α + 1 2 u t | 2 d σ .
(17)

Hence, choosing ρ 1 = 2 M ( m + 1 ) m 1 and taking into account (16), (17), and (9), we get

G ( t ) + [ 1 δ ρ 1 ] Γ 0 T | κ α + 1 2 u t | 2 d s d Γ G ( 0 ) δ 0 T u t 2 d s + δ 0 T u 2 d s + ( m + 3 ) 2 ( m + 1 ) δ 0 T u m + 1 , Γ m + 1 d s + N T σ 1 ,
(18)

where N= δ ( m 1 ) 2 | Γ | 8 M ( m + 1 ) and σ 1 = m + 1 r m 1 , |Γ| is the measure of Γ.

Choosing 0<δ< 1 ρ 1 , we may reduce the inequality (18) to

G ( T ) G ( 0 ) + δ ( m + 3 ) 2 0 T G ( s ) d s m + 8 4 δ 0 T u t 2 ( s ) d s + δ 2 ( m + 3 ) 2 0 T ( u , u t ) d s + 1 m 4 δ 0 T u 2 ( s ) d s .
(19)

By using the Cauchy-Schwarz inequality and the Poincaré inequality, we obtain

G ( t ) G ( 0 ) + δ ( m + 3 ) 2 0 T G ( s ) + δ 8 [ δ ( m + 3 ) ρ 2 ( m + 7 ) 4 ] 0 T u t 2 ( s ) d s + δ 4 [ 2 δ ρ C 0 ( m + 3 ) m + 1 ] 0 T u 2 ( s ) d s + N T σ 1 ,
(20)

where C 0 is the Poincaré constant.

Clearly, it is possible to choose

0<δ<min { 1 , 2 ( m + 7 ) ρ m + 3 , m 1 2 ρ C 0 ( m + 3 ) } .
(21)

Thus the third and the fourth terms on the right-hand side of (20) are also negative. Then

G(t)G(0)+ δ ( m + 3 ) 2 0 T G(s)ds+N T σ 1 .

We define Φ(t)=G(t). Clearly

Φ(T)Φ(0)+ δ ( m + 3 ) 2 0 T Φ(s)dsN T σ 1 .
(22)

Furthermore, we deduce from (22) that

Φ ( T ) Φ ( 0 ) e δ ( m + 3 ) T 2 σ 1 N e δ ( m + 3 ) T 2 0 T s σ 1 1 e δ ( m + 3 ) s 2 d s { Φ ( 0 ) σ 1 N ( δ ( m + 3 ) T 2 ) σ 1 Γ ( σ 1 ) } e δ ( m + 3 ) T 2 .

Since E(0) is large enough, we can choose the initial data u 0 and u 1 such that

Φ(0) σ 1 N ( δ ( m + 3 ) 2 ) σ 1 Γ( σ 1 )k>0,

then we get

Φ(T)k e δ ( m + 3 ) T 2 .
(23)

On the other hand, from the definition of Φ(t), the Cauchy-Schwarz inequality, and the Poincaré inequality, we see that

Φ(T) 1 m + 1 u m + 1 , Γ m + 1 1 δ 2 u t 2 1 δ 2 u 2 .

According to the choice of δ in (21), we get

Φ(T) ( 1 / ( m + 1 ) ) u m + 1 , Γ m + 1 .

From the formulas (23) and (22) we conclude the exponential growth of the solution in the L m + 1 (Γ) norm.

  1. (2)

    For the case 0<α< m m + 1 , we note that

    ( L κ α L T u ) ( s ) = 1 Γ ( 1 α ) 0 s ( s z ) α u ( z ) d z C 2 | s | 1 α ( m + 1 ) m ( 0 s | u ( z ) | m + 1 d z ) m m + 1 ,

where C 2 = 1 Γ ( 1 α ) ( 1 α ( m + 1 ) m ) m m + 1 with 1 α ( m + 1 ) m >0 (since 0<α< m m + 1 ). Taking this estimate into account in (12) and using the Young inequality, we find

I C 2 cos ( α 1 2 π ) ( 0 T | s | 1 ( α + 1 ) ( m + 1 ) 2 m d s ) m m + 1 ( 0 T | u | m + 1 d s ) 2 m + 1 m 1 m + 1 ( 2 ( α + 1 ) ( m + 1 ) 2 m ) m m 1 T 1 ( α + 1 ) ( m + 1 ) 2 ( m 1 ) + 2 m + 1 ( C 2 cos ( α 1 2 π ) ) m + 1 2 0 T | u | m + 1 d s .

Using this estimate in (11) and proceeding as in part (1), we may conclude.

  1. (3)

    If α= m m + 1 , we use the estimate

    I 1 cos ( α 1 2 π ) ( + | L T u | 2 d s ) 1 2 ( + ( L κ α + 1 2 L T u ) 2 ( s ) d s ) 1 2 .

In this case Lemma 2.1 is applicable with q=2, λ=1 α 2 , p=r= 2 ( m + 1 ) 2 m + 1 , we find

I C 3 Γ ( 1 α ) cos ( α 1 2 π ) ( 0 T | u | 2 d s ) 1 2 ( 0 T | u | r d s ) 1 r .

Next, by the Hölder inequality and the Young inequality, we see that

I C 3 Γ ( 1 α ) cos ( α 1 2 π ) T 2 m r m + 1 ( 0 T | u | m + 1 d s ) 2 m + 1 m 1 m + 1 T ( 2 m r ) ( m + 1 ) + 1 m + 1 ( C 3 Γ ( 1 α ) cos ( α 1 2 π ) ) m + 1 2 0 T | u | m + 1 d s .

This leads to an estimation of (12). Again the rest of the proof is similar to that in case (1). □

References

  1. 1.

    Bajlekova, EG: Fractional Evolution Equations in Banach Spaces. Dissertation, Eindhoven University of Technology, Eindhoven (2001)

    MATH  Google Scholar 

  2. 2.

    Mbodje B: Wave energy decay under fractional derivative controls. IMA J. Math. Control Inf. 2006, 237: 237-257.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    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.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    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)00023-2

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    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.2005.4.861

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    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 

  7. 7.

    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 1998

    Google Scholar 

  8. 8.

    Mbodje B, Montsney G: Boundary fractional derivative control of the wave equation. IEEE Trans. Autom. Control 1995, 40(2):378-382. 10.1109/9.341815

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    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. 10.1080/00207729408949312

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    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.1994.381303

    Google Scholar 

  11. 11.

    Kirane M, Tatar N: Exponential growth for fractionally damped wave equation. Z. Anal. Anwend. 2003, 22(1):167-177. 10.4171/ZAA/1137

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Tatar N: A blow up result for a fractionally damped wave equation. Nonlinear Differ. Equ. Appl. 2005, 12: 215-226. 10.1007/s00030-005-0015-6

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    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.

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    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/961

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Aassila M: Nonexistence of global solutions of a hyperbolic problem. Math. Comput. Model. 2001, 34: 761-769. 10.1016/S0895-7177(01)00097-8

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    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/S0363012998344981

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    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.2003.12.057

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    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.

    MathSciNet  MATH  Google Scholar 

  19. 19.

    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.2010.10.012

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    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.2011.04.030

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Hormander L: The Analysis of Linear Partial Differential Operators. I. Springer, Berlin; 1983.

    MATH  Google Scholar 

  22. 22.

    Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton; 1993.

    MATH  Google Scholar 

  23. 23.

    Lions JL, Magenes E: Non-Homogeneous Boundary Value Problem and Applications. Springer, Berlin; 1983.

    MATH  Google Scholar 

  24. 24.

    Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.

    MATH  Google Scholar 

  25. 25.

    Greenberg G, Londen SO, Staffans O: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge; 1990.

    Book  MATH  Google Scholar 

Download references

Acknowledgements

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Huiya Dai.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dai, H., Zhang, H. Exponential growth for wave equation with fractional boundary dissipation and boundary source term. Bound Value Probl 2014, 138 (2014). https://doi.org/10.1186/s13661-014-0138-y

Download citation

Keywords

  • exponential growth
  • fractional boundary dissipation
  • wave equation