Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition
Boundary Value Problemsvolume 2013, Article number: 166 (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.
The aim of this article is to study the energy decay and blow-up of a solution of the following Kirchhoff-type equation with nonlinear dynamic boundary condition:
here , , , are positive constants and .
This problem is based on the equation
which was proposed by Woinowsky-Krieger [1, 2] as a model for a vibrating beam with hinged ends, where is the lateral displacement at the space coordinate x and time t. Equation (6) was studied by many authors such as Dickey , Ball Rivera , Tucsnak , Kouemou Patchen , Aassila , Oliveira and Lima ; Wu and Tsai  considered the following beam equation:
They obtained the existence and uniqueness, as well as decay estimates, of global solutions and blow-up of solutions for the initial boundary value problem of equation (7) through various approaches and assumptive conditions. Feireisl  and Fitzgibbon et al.  showed the existence of a global attractor and an inertial manifold of equation (6) with damping . Ma  studied the existence and decay rates for the solution of equation (6) with nonlinear boundary conditions
Pazoto and Menzala  were concerned with equation (6) with rotational inertia term and nonlinear boundary condition (8). Santos et al.  established the existence and exponential decay of the Kirchhoff systems with nonlocal boundary condition. Guedda and Labani  gave the sufficient condition of the blow-up of the solution to equation (7) with and dynamic boundary condition. As the related problem, we mention the following:
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 .
If , then(9)
If and , then(10)
Proof Since , we have
Take , we get the first inequality of (9). Integrating the above inequality over , we get the second part of (9). From the following equation
and the Cauchy inequality, we can get the result of (10) with the help of (9). □
Lemma 2.2 
Let be a non-increasing and nonnegative function on such that
where C, ω are positive constants depending on and other known qualities.
Lemma 2.3 
Suppose that a positive, twice-differentiable function satisfies on the inequality
where , then there is a such that as .
A solution u of problem (1)-(5) means that there exists such that
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 blow-up 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, is a non-increasing function on . Moreover, we have
We can now define a stable set and an unstable set 
3 Energy decay of the solution
In order to get the energy decay of the solution, we introduce the following set:
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 .
Lemma 3.2 Under the condition of Lemma 3.1 and , then, for ,
Proof By (14) and (18), we have
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 .
By (24), we have
then (21) holds since . Furthermore, we have
So (22) holds. Similar to (25), the above equality becomes
so (23) holds. □
Theorem 3.3 Let , , and u be the solution of problem (1)-(5), then there exist two positive constants l and θ independent of t such that
Proof From (16), we have
Now, for the above estimate and the mean value theorem, we choose and such that, for ,
Multiplying equation (1) by u and integrating over , by the boundary conditions (2) and (3), we have
Now, we estimate the terms of the right-hand 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 is non-increasing, we arrive at
By the Young inequality, (9), (10), (23), (26) and the fact that is non-increasing, we obtain
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 is non-increasing, we choose such that
Then, using (26), , and the fact that is non-increasing, we have
Since is non-increasing, combining this with (40), (38) and (26), we have
Choosing sufficiently small, (41) leads to
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 .
Lemma 4.1 Suppose , , then
Proof Since is non-increasing, and , then for . Similar to the proof of Lemma 3.2, we have
where . Note that has the maximum at and the maximum value is . It is easy to verify that is increasing for , decreasing in and as . Therefore, since , there exists such that . By (44), we have , which implies . We claim that
Otherwise, we suppose that for some and by the continuity of , we can choose such that .
Again the use of (44) leads to
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.
where , , β are positive constants which will be fixed later (see Levine ). Then one finds
By (17) and (14), we have
Taking and noticing , we get
By the Hölder inequality, we have
Then, by (48), (51), (52) and the Cauchy-Schwarz inequality, we arrive at
Take sufficiently large such that
Noticing , by Lemma 2.3, we get the result. □
Theorem 4.3 Suppose that is the local solution of problem (1)-(5), , and that either of the following conditions is satisfied:
and (or );
then the solution u blows up at some finite time.
Proof (i) For , similar to the proof of Theorem 4.2, we take in (51), then (53) holds. Since , , then the result holds by Lemma 2.3.
For the case of , from (48), (49), (50) and (14), we get(55)
By Lemma 4.1,
Combining (55) with (56), and (17), we get
Take , then , since and , then (57) can be rewritten
The remainder of the proof is the same as the proof of Theorem 4.2. □
Woinowsky-Krieger S: The effect of axial force on the vibration of hinged bars. J. Appl. Mech. 1980, 17: 35–36.
Ball J: Stability theory for an extensible beam. J. Differ. Equ. 1973, 14: 58–66.
Dickey KW: Infinite systems of nonlinear oscillation equations with linear damping. SIAM J. Appl. Math. 1970, 7: 208–214.
Munoz Rivera JE: Global stabilization and regularizing properties on a class of nonlinear evolution equation. J. Differ. Equ. 1996, 128: 103–124. 10.1006/jdeq.1996.0091
Tucsnak M: Semi-internal stabilization for a nonlinear Euler-Bernoulli equation. Math. Methods Appl. Sci. 1996, 9: 897–907.
Patcheu SK: On a global solution and asymptotic behavior for the generalized damped extensible beam equation. J. Differ. Equ. 1997, 135(2):123–138.
Aassila M: Decay estimate for a quasi-linear wave equation of Kirchhoff type. Adv. Math. Sci. Appl. 1999, 9(1):371–381.
Oliveira ML, Lima OA: Exponential decay of the solutions of the beams system. Nonlinear Anal. 2000, 42: 1271–1291. 10.1016/S0362-546X(99)00155-8
Wu ST, Tsai LY: Existence and nonexistence of global solutions for a nonlinear wave equation. Taiwan. J. Math. 2009, 13B(6):2069–2091.
Feireisl F: Exponential attractor for non-autonomous systems long-time behavior of vibrating beams. Math. Methods Appl. Sci. 1992, 15: 287–297. 10.1002/mma.1670150406
Fitzgibbon WE, Parrott M, You YC: Global dynamics of coupled systems modelling non-planar beam motion. Lectures Notes in Pure and Appl. Math. Ins. 168. In Evolution Equation. Edited by: Ferreyra G, Goldstein GK, Neubrander F. Marcel Dekker, New York; 1995:187–189.
Ma TF: Boundary stabilization for a nonlinear beam on elastic bearings. Math. Methods Appl. Sci. 2001, 24: 583–594. 10.1002/mma.230
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. 2005, 6: 1–28.
Guedda M, Labani H: Nonexistence of global solutions to a class of nonlinear wave equations with dynamic boundary conditions. Bull. Belg. Math. Sci. 2002, 9: 39–46.
Autuori G, Pucci P, Salvaton MC: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal., Real World Appl. 2009, 10: 889–909. 10.1016/j.nonrwa.2007.11.011
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. 2010, 36: 566–578.
Nakao M: An attractor for a nonlinear dissipative wave equation of Kirchhoff type. J. Math. Anal. Appl. 2009, 353: 652–659. 10.1016/j.jmaa.2008.09.010
Li FC: Global existence and blow-up of solutions for higher-order Kirchhoff-type equation with nonlinear dissipation. Appl. Math. Lett. 2004, 17: 1409–1414. 10.1016/j.am1.2003.07.014
Grobbelaar-van Dalsen M: On the initial-boundary-value problem for the extensible beam with attached load. Math. Methods Appl. Sci. 1996, 19: 943–957. 10.1002/(SICI)1099-1476(199608)19:12<943::AID-MMA804>3.0.CO;2-F
Grobbelaar-van Dalsen M, Van der Merwe A: Boundary stabilization for the extensible beam with attached load. Math. Models Methods Appl. Sci. 1999, 9: 379–394. 10.1142/S0218202599000191
Grobbelaar-van Dalsen M: On the solvability of the boundary-value problem for the elastic beam with attached load. Math. Models Methods Appl. Sci. 1994, 4: 89–105. 10.1142/S0218202594000066
Littman W, Markus L: Stabilization of a hybrid system of elasticity by feedback boundary damping. Ann. Math. Pures Appl. 1988, 152: 281–330. 10.1007/BF01766154
Andrews KT, Kuttler KL, Shillor M: Second order evolution equation with dynamic boundary conditions. J. Math. Anal. Appl. 1996, 197: 781–795. 10.1006/jmaa.1996.0053
Conrad F, Morgul O: On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 1998, 36: 1962–1966. 10.1137/S0363012996302366
Rao BP: Uniform stabilization of a hybrid system of elasticity. SIAM J. Control Optim. 1995, 33: 440–454. 10.1137/S0363012992239879
Park JY, Park SH: Solution for a hyperbolic system with boundary differential inclusion and nonlinear second-order boundary damping. Electron. J. Differ. Equ. 2003, 80: 1–7.
Doronin GG, Larkin NA: Global solvability for the quasi-linear damped wave equation with nonlinear second-order boundary condition. Nonlinear Anal. 2002, 8: 1119–1134.
Gerbi S, Said-Houari B: Local existence and exponential growth for a semi-linear damped wave equation with dynamical boundary conditions. Adv. Differ. Equ. 2008, 13: 1051–1060.
Autuori G, Pucci P: Kirchhoff system with dynamic boundary conditions. Nonlinear Anal. 2010, 73: 1952–1965. 10.1016/j.na.2010.05.024
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. 1999, 239: 213–226. 10.1006/jmaa.1999.6528
Payne L, Sattinger D: Saddle points and instability on nonlinear hyperbolic equations. Isr. J. Math. 1973, 22: 273–303.
Levine HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 1974, 5: 138–146. 10.1137/0505015
Nakao M, Ono K: Global existence to the Cauchy problem of the semi-linear evolution equations with a nonlinear dissipation. Funkc. Ekvacioj 1995, 38: 417–431.
Vitillaro E: A potential well method for the wave equation with nonlinear source and boundary damping terms. Glasg. Math. J. 2002, 44: 375–395. 10.1017/S0017089502030045
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).
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.