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Exponential growth for wave equation with fractional boundary dissipation and boundary source term
Boundary Value Problems volume 2014, Article number: 138 (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.
In this paper we consider the unboundedness of the classical energy for the following problem:
where Ω is a bounded domain in with smooth boundary Γ such that , is a constant. The initial data and are given function, denotes the outward normal derivative. The notation stands for the Caputo fractional derivative of order α with respect to the time variable , . It is defined as follows:
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.
Throughout this paper, we denote by , the usual Sobolev space, and
Let us define
It is easily seen that
Observe that 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
Hence is uniformly bounded by , this is
Let, , and, thenwith. Also the mapping fromintois continuous.
In the subspace
of the Sobolev space, there exists a constantsuch that
If we denoteby, then we have
3 Exponential growth of the solution
Let us define
where is a small constant to be determined later. Multiplying (1) by and then integrating over Ω, we get
For simplicity, we denote
Then, using the definition of and integrating (8) over for t, we may write
Next, we estimate, for a fixed ,
where is as in Lemma 2.3. Then we can easily see that
where we have used the Parseval theorem  and denoted by the usual Fourier transform of f. Then, using Lemma 2.3, the Cauchy-Schwarz inequality, and the Young inequality, we obtain for
For the last term in (11), we have
here we have used the Hölder inequality and the following inequality , Theorem 16.5.1]:
Next we consider three cases.
If , then the Lemma 2.1 with
where depends only m and α.
Since , it is easily to see that
where . Analogously,
where and , is the measure of Γ.
Choosing , we may reduce the inequality (18) to
By using the Cauchy-Schwarz inequality and the Poincaré inequality, we obtain
where is the Poincaré constant.
Clearly, it is possible to choose
Thus the third and the fourth terms on the right-hand side of (20) are also negative. Then
We define . Clearly
Furthermore, we deduce from (22) that
Since is large enough, we can choose the initial data and such that
then we get
On the other hand, from the definition of , the Cauchy-Schwarz inequality, and the Poincaré inequality, we see that
According to the choice of δ in (21), we get
For the case , we note that
where with (since ). Taking this estimate into account in (12) and using the Young inequality, we find
Using this estimate in (11) and proceeding as in part (1), we may conclude.
If , we use the estimate
In this case Lemma 2.1 is applicable with , , , we find
Next, by the Hölder inequality and the Young inequality, we see that
This leads to an estimation of (12). Again the rest of the proof is similar to that in case (1). □
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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).
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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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
- exponential growth
- fractional boundary dissipation
- wave equation