Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation
© Ye; licensee Springer 2013
Received: 25 August 2012
Accepted: 9 April 2013
Published: 22 April 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.
Keywordsnonlinear Kirchhoff-type equation initial boundary value problem stable set energy decay estimate
with is a constant.
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 .
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.
for , , and is the total energy of the initial data.
Lemma 2.2 
then , , if , where C is positive constants independent of .
Therefore, is a nonincreasing function on t. □
In order to prove the existence of global solutions of the problem (1.1)-(1.3), we need the following lemma.
then , for each .
for all .
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 .
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.
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 .
Let , then we obtain .
Since the initial data is such that , satisfies , .
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).
where is a constant depending on initial energy .
The proof of Theorem 4.1 is thus finished. □
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).
- Arosio A, Garavaldi S: On the mildly degenerate Kirchhoff string. Math. Methods Appl. Sci. 1991, 14: 177–195.MathSciNetView ArticleGoogle Scholar
- Crippa HR: On local solutions of some mildly degenerate hyperbolic equations. Nonlinear Anal. 1993, 21: 565–574. 10.1016/0362-546X(93)90001-9MathSciNetView ArticleGoogle Scholar
- Ebihara Y, Medeiros LA, Miranda MM: Local solutions for nonlinear degenerate hyperbolic equation. Nonlinear Anal. 1986, 10: 27–40. 10.1016/0362-546X(86)90009-XMathSciNetView ArticleGoogle Scholar
- Meideiros LA, Miranda M: Solutions for the equations of nonlinear vibrations in Sobolev spaces of fractionary order. Comput. Appl. Math. 1987, 6: 257–276.Google Scholar
- Yamada Y: Some nonlinear degenerate wave equations. Nonlinear Anal. 1987, 11: 1155–1168. 10.1016/0362-546X(87)90004-6MathSciNetView ArticleGoogle Scholar
- Yamazaki T: On local solution of some quasilinear degenerate hyperbolic equations. Funkc. Ekvacioj 1988, 31: 439–457.Google Scholar
- Nishihara K, Yamada Y: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkc. Ekvacioj 1990, 33: 151–159.MathSciNetGoogle Scholar
- Aassila M, Benaissa A: Existence globale et comportement asymptotique des solutions des équations de Kirchhoff moyennement dégénérées avec un terme nonlinear dissipatif. Funkc. Ekvacioj 2001, 44: 309–333.MathSciNetGoogle Scholar
- D’Ancona P, Spagnolo S: Nolinear perturbations of the Kirchhoff equation. Commun. Pure Appl. Math. 1994, 47: 1005–1029. 10.1002/cpa.3160470705MathSciNetView ArticleGoogle Scholar
- De Brito EH: The damped elastic stretched string equation generalized: existence uniqueness, regularity and stability. Appl. Anal. 1982, 13: 219–233. 10.1080/00036818208839392MathSciNetView ArticleGoogle Scholar
- De Brito EH: Decay estimates for the generalized damped extensible string and beam equation. Nonlinear Anal. 1984, 8: 1489–1496. 10.1016/0362-546X(84)90059-2MathSciNetView ArticleGoogle Scholar
- Nishihara K: Global existence and asymptotic behavior of the solution of some quasilinear hyperbolic equation with linear damping. Funkc. Ekvacioj 1989, 32: 343–355.MathSciNetGoogle Scholar
- Yamada Y: On some quasilinear wave equations with dissipative terms. Nagoya Math. J. 1982, 87: 17–39.MathSciNetGoogle Scholar
- Ghisi M, Gobbino M: Global existence and asymptotic behaviour for a mildly degenerate dissipative hyperbolic equation of Kirchhoff type. Asymptot. Anal. 2004, 40: 25–36.MathSciNetGoogle Scholar
- Hosoya M, Yamada Y: On some nonlinear wave equations II: global existence and energy decay of solutions. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 1991, 38: 239–250.MathSciNetGoogle Scholar
- Sattinger DH: On global solutions for nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 1968, 30: 148–172.MathSciNetView ArticleGoogle Scholar
- Payne LE, Sattinger DH: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 1975, 22: 273–303. 10.1007/BF02761595MathSciNetView ArticleGoogle Scholar
- Komornik V: Exact Controllability and Stabilization, the Multiplier Method. Masson, Paris; 1994.Google Scholar
- Ono K: Global existence and decay properties of solutions for some mildly degenerate nonlinear dissipative Kirchhoff strings. Funkc. Ekvacioj 1997, 40: 255–270.Google Scholar
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