Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition
© Zhang et al.; licensee Springer. 2013
Received: 16 November 2012
Accepted: 27 June 2013
Published: 12 July 2013
The energy decay and blow-up of a solution for a Kirchhoff equation with dynamic boundary condition are considered. With the help of Nakao’s inequality and a stable set, the energy decay of the solution is given. By the convexity inequality lemma and an unstable set, the sufficient condition of blow-up of the solution with negative and small positive initial energy are obtained, respectively.
here , , , are positive constants and .
When and , problem (1)-(5) comes from the reference [20–22]. In this case, the model describes the weakly damped vibrations of an extensible beam whose ends are a fixed distance apart if one end is hinged while a load is attached to the other end . One can find many references on problem (1)-(5) with and , for example, Littman and Markus , Andrews et al. , Conrad and Morgul , Rao .
Dalsen [21, 22] showed the exponential stability of problem (1)-(5) with and . Park et al.  discussed the existence of the solution of the Kirchhoff equation with dynamic boundary conditions and boundary differential inclusion. Doronin and Larkin  and Gerbi and Said-Houari  were concerned with the wave equation with dynamic boundary conditions. Recently, Autuori and Pucci  studied the global nonexistence of solutions of the p-Kirchhoff system with dynamic boundary condition.
In this paper, we use the idea of references  to get the energy decay and blow-up of the solution for problem (1)-(5). We construct a stable set and an unstable set, which is similar to . By the help of Nakao’s inequality, combining it with the stable set, we get the decay estimate. We find that the set of initial data such that the solution of problem (1)-(5) is decay, is smaller than the potential well in . The blow-up properties of the solution of problem (1)-(5) with small positive initial energy and negative initial energy are obtained by using the convexity lemma . These results are different from the results in [29, 30].
2 Assumptions and preliminaries
In this section, we give some preliminaries which are used throughout this work.
We use the standard space and the Sobolev space , with their usual scalar products and norms. Especially, denotes the norm of and the norms .
We denote .
- (1)If , then(9)
- (2)If and , then(10)
and the Cauchy inequality, we can get the result of (10) with the help of (9). □
Lemma 2.2 
where C, ω are positive constants depending on and other known qualities.
Lemma 2.3 
where , then there is a such that as .
for all .
3 Energy decay of the solution
where , . Obviously, .
Adapting the idea of Vitillaro , we have the following lemma.
Lemma 3.1 Suppose that u is the solution of (1)-(5), , and , then , for .
where . Note that has the maximum at and the maximum value . We see that is increasing in , decreasing in and as . Since , , then for any , so .
so (23) holds. □
then, applying Lemma 2.2, we obtain the energy decay. □
4 Blow-up property
In this section, we show that the solution of problem (1)-(5) blows up in finite time if .
Otherwise, we suppose that for some and by the continuity of , we can choose such that .
This is impossible since for all . Hence, (45) holds. Furthermore, (43) is established since . □
Theorem 4.2 Suppose that u is the local solution of problem (1)-(5), , , then the solution u blows up at some finite time.
Noticing , by Lemma 2.3, we get the result. □
and (or );
then the solution u blows up at some finite time.
- (ii)For the case of , from (48), (49), (50) and (14), we get(55)
The remainder of the proof is the same as the proof of Theorem 4.2. □
We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11171311) and the Key Science Foundation of Henan University of Technology (09XZD009).
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