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Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation
Journal of Inequalities and Applicationsvolume 2013, Article number: 195 (2013)
This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equation with dissipative term
in a bounded domain, where and are constants. We obtain the global existence of solutions by constructing a stable set in and show the energy decay estimate by applying a lemma of Komornik.
In this paper, we investigate the existence and asymptotic stability of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with nonlinear dissipative term in a bounded domain
where Ω is a bounded domain in with a smooth boundary ∂ Ω, and are constants, is a -class function on satisfying
with is a constant.
If is an interval of the real line, equation (1.1) describes a small amplitude vibration of an elastic string with fixed endpoints. The original equation is
where L is the rest length, E is the Young modulus, ρ is the mass density, h is the cross-section area, is the initial axial tension, δ is the resistance modulus and f is a nonlinear perturbation effect.
When , , and (the mildly degenerate case), the local existence of solutions in Sobolev space was investigated by many author [1–6]. Concerning a global existence of solutions for mildly degenerate Kirchhoff equations, it is natural to add a dissipative term or .
For , , , , , the problem (1.1)-(1.3) was treated by Nishihara and Yamada . They proved the existence and uniqueness of a global solution for small data with and the polynomial decay of the solution. Aassila and Benaissa  extended the global existence part of  to the case where with and the case of nonlinear dissipative term case ().
In the case , for large β and , D’Ancona and Spagnolo  proved that if are small, then problem (1.1)-(1.3) has a global solution. The nondegenerate case with , and was considered by De Brito, Yamada and Nishihara [10–13], they proved that for small initial data there exists a unique global solution of (1.1)-(1.3) that decays exponentially as .
When , Ghisi and Gobbino  proved the existence and uniqueness of a global solution of the problem (1.1)-(1.3) for small initial data with and the asymptotic behavior in as , where either or .
The case has been considered by Hosoya and Yamada  under the following condition:
They proved that, if the initial datas are small enough, the problem (1.1)-(1.3) has a global solution which decays exponentially as .
In this paper, we prove the global existence for the problem (1.1)-(1.3) by applying the potential well theory introduced by Sattinger  and Payne and Sattinger . Meanwhile, we obtain the asymptotic stability of global solutions by use of the lemma of Komornik .
We adopt the usual notation and convention. Let denote the Sobolev space with the norm , denotes the closure in of . For simplicity of notation, hereafter we denote by the Lebesgue space norm, denotes norm and we write equivalent norm instead of norm . Moreover, M denotes various positive constants depending on the known constants and it may be difference in each appearance.
This paper is organized as follows: In the next section, we will give some preliminaries. Then in Section 3, we state the main results and give their proof.
In order to state and prove our main results, we first define the following functionals:
for . Then we define the stable set S by
We denote the total energy functional associated with (1.1)-(1.3) by
for , , and is the total energy of the initial data.
Lemma 2.1 Let q be a number with , and , . Then there is a constant C depending on Ω and q such that
Lemma 2.2 
Let be a nonincreasing function and assume that there are two constants and such that
then , , if , where C is positive constants independent of .
Lemma 2.3 Let be a solutions to the problem (1.1)-(1.3). Then is a nonincreasing function for and
Proof By multiplying equation (1.1) by and integrating over Ω, we get
Therefore, is a nonincreasing function on t. □
Theorem 2.1 Suppose that satisfy
and let . Then there exists such that the problem (1.1)-(1.3) has a unique local solution in the class
In order to prove the existence of global solutions of the problem (1.1)-(1.3), we need the following lemma.
Lemma 2.4 Supposed that (2.3) holds, If , such that
then , for each .
Proof Assume that there exists a number such that on and . Then we have
Since on , it holds that
we have from that
we conclude from (1.4) and (2.1) that
Therefore, we obtain from (2.8), (2.9) and (2.10) that
By exploiting Lemma 2.1, (2.6) and (2.11), we easily arrive at
for all .
Therefore, we obtain
which contradicts (2.7). Thus, we conclude that on . □
3 The global existence and nonexistence
Theorem 3.1 Suppose that (2.3) and (2.4) hold, and is a local solution of problem (1.1)-(1.3) on . If and satisfy (2.6), then is a global solution of the problem (1.1)-(1.3).
Proof It suffices to show that is bounded independently of t.
Under the hypotheses in Theorem 3.1, we get from Lemma 2.4 that on . So the formula (2.8) holds on .
Therefore, we have from (2.8) that
Hence, we get
The above inequality and the continuation principle lead to the global existence of the solution, that is, . Thus, the solution is a global solution of the problem (1.1)-(1.3). □
Now we employ the analysis method to discuss the solution of the problem (1.1)-(1.3) occurs blow-up in finite time. Our result reads as follows.
Theorem 3.2 Assume that (i) , if ; (ii) , if . If and such that
with is a positive Sobolev constant. Then the solution of the problem (1.1)-(1.3) does not exist globally in time.
Proof On the contrary, under the conditions in Theorem 3.2, suppose that is a global solution of the problem (1.1)-(1.3); then by Lemma 2.1, it is well known that there exists a constant depending only n, β such that for all .
From the above inequality, we conclude that
It follows from (1.4), (2.1) and (3.2) that
We denote the right side of (3.3) by , then
By (3.4), we obtain
Let , then we obtain .
As , we have
Consequently, the function has a single maximum value at , where
Since the initial data is such that , satisfies , .
Therefore, we have from Lemma 2.3 that
At the same time, by (3.3) and (3.4) it is evident that there can be no time for which
Hence, we have also for all from the continuity of and .
According to the above contradiction we know that the global solution of the problem (1.1)-(1.3) does not exist, i.e., the solution blows up in some finite time.
This completes the proof of Theorem 3.2. □
4 Energy decay estimate
The following theorem shows the asymptotic behavior of global solutions of the problem (1.1)-(1.3).
Theorem 4.1 If the hypotheses in Theorem 3.2 are valid, then the global solutions of the problem (1.1)-(1.3) has the following asymptotic property:
where is a constant depending on initial energy .
Proof Multiplying by on both sides of the equation (1.1) and integrating over , we obtain that
So, substituting the formula (4.2) into the right-hand side of (4.1), we get that
We obtain from (2.12) and (2.11) that
We derive from (1.4) that
It follows from (4.3), (4.4) and (4.5) that
We have from Lemma 2.1 and (3.1) that
similarly, we have
Substituting the estimates (4.7) and (4.8) into (4.6), we conclude that
We get from Young inequality and Lemma 2.3 that
From Young inequality, Lemma 2.1, Lemma 2.3 and (2.11), We receive that
Choosing small enough and , such that
then, substituting (4.10) and (4.11) into (4.9), we get
Therefore, we have from Lemma 2.2 that
The proof of Theorem 4.1 is thus finished. □
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This work was supported by the Natural Science Foundation of China (No. 61273016), the Natural Science Foundation of Zhejiang Province (No. Y6100016), the Middle-aged Academic Leader of Zhejiang University of Science and Technology (2008-2012), Interdisciplinary Pre-research Project of Zhejiang University of Science and Technology (2010-2012) and Zhejiang province universities scientific research key project (Z201017584).
The author declares that he has no competing interests.