Abstract
The energy decay and blowup 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 blowup of the solution with negative and small positive initial energy are obtained, respectively.
1 Introduction
The aim of this article is to study the energy decay and blowup of a solution of the following Kirchhofftype equation with nonlinear dynamic boundary condition:
here
This problem is based on the equation
which was proposed by WoinowskyKrieger [1,2] as a model for a vibrating beam with hinged ends, where
They obtained the existence and uniqueness, as well as decay estimates, of global
solutions and blowup of solutions for the initial boundary value problem of equation
(7) through various approaches and assumptive conditions. Feireisl [10] and Fitzgibbon et al.[11] showed the existence of a global attractor and an inertial manifold of equation (6)
with damping
Pazoto and Menzala [13] were concerned with equation (6) with rotational inertia term
we refer the reader to [1619].
When
Dalsen [21,22] showed the exponential stability of problem (1)(5) with
In this paper, we use the idea of references [31] to get the energy decay and blowup of the solution for problem (1)(5). We construct a stable set and an unstable set, which is similar to [32]. 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 [32]. The blowup properties of the solution of problem (1)(5) with small positive initial energy and negative initial energy are obtained by using the convexity lemma [33]. 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
We denote
Lemma 2.1
(1) If
(2) If
Proof Since
Take
and the Cauchy inequality, we can get the result of (10) with the help of (9). □
Lemma 2.2[34]
Let
then
whereC, ωare positive constants depending on
Lemma 2.3[33]
Suppose that a positive, twicedifferentiable function
where
A solution u of problem (1)(5) means that there exists
and it satisfies the following identity
for all
In this paper, we always assume that a local solution exists for problem (1)(5). In order to study the energy decay or the blowup phenomenon of the solution of problem (1)(5), we define the energy of the solution u of problem (1)(5) by
The initial energy is defined by
Then, after some simple computation, we have
That is to say,
We denote
We can now define a stable set and an unstable set [31]
3 Energy decay of the solution
In order to get the energy decay of the solution, we introduce the following set:
where
Adapting the idea of Vitillaro [35], we have the following lemma.
Lemma 3.1Suppose thatuis the solution of (1)(5),
Lemma 3.2Under the condition of Lemma 3.1 and
Proof By (14) and (18), we have
where
By (24), we have
then (21) holds since
So (22) holds. Similar to (25), the above equality becomes
so (23) holds. □
Theorem 3.3Let
Proof From (16), we have
Now, for the above estimate and the mean value theorem, we choose
Multiplying equation (1) by u and integrating over
Now, we estimate the terms of the righthand side of (29). By (10), (23), (27) and the Young inequality, we have
It follows from (9), (10), (23), (28) and the Young inequality that
From (26), we get
From the Young inequality, (10), (23), (26) and from the fact that
By the Young inequality, (9), (10), (23), (26) and the fact that
Substituting (30)(34) into (29), we get the estimate
On the other hand, it follows from (22) that
Then we have
Therefore, by (37), (35) and (26), we arrive at
Since
Then, using (26),
Since
Choosing
then, applying Lemma 2.2, we obtain the energy decay. □
4 Blowup property
In this section, we show that the solution of problem (1)(5) blows up in finite time
if
Lemma 4.1Suppose
Proof Since
where
Otherwise, we suppose that
Again the use of (44) leads to
This is impossible since
Theorem 4.2Suppose thatuis the local solution of problem (1)(5),
Proof Let
where
By (17) and (14), we have
Taking
By the Hölder inequality, we have
Denote
Then, by (48), (51), (52) and the CauchySchwarz inequality, we arrive at
Take
Noticing
Theorem 4.3Suppose that
(i)
(ii)
then the solutionublows up at some finite time.
Proof (i) For
(ii) For the case of
By Lemma 4.1,
Combining (55) with (56),
Take
The remainder of the proof is the same as the proof of Theorem 4.2. □
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.
Acknowledgements
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).
References

WoinowskyKrieger, S: The effect of axial force on the vibration of hinged bars. J. Appl. Mech.. 17, 35–36 (1980)

Ball, J: Stability theory for an extensible beam. J. Differ. Equ.. 14, 58–66 (1973)

Dickey, KW: Infinite systems of nonlinear oscillation equations with linear damping. SIAM J. Appl. Math.. 7, 208–214 (1970)

Munoz Rivera, JE: Global stabilization and regularizing properties on a class of nonlinear evolution equation. J. Differ. Equ.. 128, 103–124 (1996). Publisher Full Text

Tucsnak, M: Semiinternal stabilization for a nonlinear EulerBernoulli equation. Math. Methods Appl. Sci.. 9, 897–907 (1996)

Patcheu, SK: On a global solution and asymptotic behavior for the generalized damped extensible beam equation. J. Differ. Equ.. 135(2), 123–138 (1997). PubMed Abstract

Aassila, M: Decay estimate for a quasilinear wave equation of Kirchhoff type. Adv. Math. Sci. Appl.. 9(1), 371–381 (1999)

Oliveira, ML, Lima, OA: Exponential decay of the solutions of the beams system. Nonlinear Anal.. 42, 1271–1291 (2000). Publisher Full Text

Wu, ST, Tsai, LY: Existence and nonexistence of global solutions for a nonlinear wave equation. Taiwan. J. Math.. 13B(6), 2069–2091 (2009)

Feireisl, F: Exponential attractor for nonautonomous systems longtime behavior of vibrating beams. Math. Methods Appl. Sci.. 15, 287–297 (1992). Publisher Full Text

Fitzgibbon, WE, Parrott, M, You, YC: Global dynamics of coupled systems modelling nonplanar beam motion. In: Ferreyra G, Goldstein GK, Neubrander F (eds.) Evolution Equation, pp. 187–189. Marcel Dekker, New York (1995)

Ma, TF: Boundary stabilization for a nonlinear beam on elastic bearings. Math. Methods Appl. Sci.. 24, 583–594 (2001). Publisher Full Text

Pazoto, AF, Menzala, GP: Uniform rates of decay of a nonlinear beam with boundary dissipation. Report of LNCC/CNPq (Brazil), no 34/97, August 1997.

Santos, ML, Rocha, MPC, Pereira, DC: Solvability for a nonlinear coupled system of Kirchhoff type for the beam equations with nonlocal boundary conditions. Electron. J. Qual. Theory Differ. Equ.. 6, 1–28 (2005)

Guedda, M, Labani, H: Nonexistence of global solutions to a class of nonlinear wave equations with dynamic boundary conditions. Bull. Belg. Math. Sci.. 9, 39–46 (2002)

Autuori, G, Pucci, P, Salvaton, MC: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal., Real World Appl.. 10, 889–909 (2009). Publisher Full Text

Tawiguchi, T: Existence and asymptotic behavior of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms. J. Math. Anal. Appl.. 36, 566–578 (2010)

Nakao, M: An attractor for a nonlinear dissipative wave equation of Kirchhoff type. J. Math. Anal. Appl.. 353, 652–659 (2009). Publisher Full Text

Li, FC: Global existence and blowup of solutions for higherorder Kirchhofftype equation with nonlinear dissipation. Appl. Math. Lett.. 17, 1409–1414 (2004). Publisher Full Text

Grobbelaarvan Dalsen, M: On the initialboundaryvalue problem for the extensible beam with attached load. Math. Methods Appl. Sci.. 19, 943–957 (1996). Publisher Full Text

Grobbelaarvan Dalsen, M, Van der Merwe, A: Boundary stabilization for the extensible beam with attached load. Math. Models Methods Appl. Sci.. 9, 379–394 (1999). Publisher Full Text

Grobbelaarvan Dalsen, M: On the solvability of the boundaryvalue problem for the elastic beam with attached load. Math. Models Methods Appl. Sci.. 4, 89–105 (1994). Publisher Full Text

Littman, W, Markus, L: Stabilization of a hybrid system of elasticity by feedback boundary damping. Ann. Math. Pures Appl.. 152, 281–330 (1988). Publisher Full Text

Andrews, KT, Kuttler, KL, Shillor, M: Second order evolution equation with dynamic boundary conditions. J. Math. Anal. Appl.. 197, 781–795 (1996). Publisher Full Text

Conrad, F, Morgul, O: On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim.. 36, 1962–1966 (1998). Publisher Full Text

Rao, BP: Uniform stabilization of a hybrid system of elasticity. SIAM J. Control Optim.. 33, 440–454 (1995). Publisher Full Text

Park, JY, Park, SH: Solution for a hyperbolic system with boundary differential inclusion and nonlinear secondorder boundary damping. Electron. J. Differ. Equ.. 80, 1–7 (2003)

Doronin, GG, Larkin, NA: Global solvability for the quasilinear damped wave equation with nonlinear secondorder boundary condition. Nonlinear Anal.. 8, 1119–1134 (2002)

Gerbi, S, SaidHouari, B: Local existence and exponential growth for a semilinear damped wave equation with dynamical boundary conditions. Adv. Differ. Equ.. 13, 1051–1060 (2008)

Autuori, G, Pucci, P: Kirchhoff system with dynamic boundary conditions. Nonlinear Anal.. 73, 1952–1965 (2010). Publisher Full Text

Todorova, G: Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. J. Math. Anal. Appl.. 239, 213–226 (1999). Publisher Full Text

Payne, L, Sattinger, D: Saddle points and instability on nonlinear hyperbolic equations. Isr. J. Math.. 22, 273–303 (1973)

Levine, HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal.. 5, 138–146 (1974). Publisher Full Text

Nakao, M, Ono, K: Global existence to the Cauchy problem of the semilinear evolution equations with a nonlinear dissipation. Funkc. Ekvacioj. 38, 417–431 (1995)

Vitillaro, E: A potential well method for the wave equation with nonlinear source and boundary damping terms. Glasg. Math. J.. 44, 375–395 (2002). Publisher Full Text