# Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition

- Hongwei Zhang
^{1}Email author, - Changshun Hou
^{1}and - Qingying Hu
^{1}

**2013**:166

**DOI: **10.1186/1687-2770-2013-166

© Zhang et al.; licensee Springer. 2013

**Received: **16 November 2012

**Accepted: **27 June 2013

**Published: **12 July 2013

## Abstract

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.

## 1 Introduction

here $f(s)=|s{|}^{p-2}s$, $M(s)=1+{s}^{m}$, $p>2$, $m\ge 1$ are positive constants and ${\parallel {u}_{x}\parallel}^{2}={\int}_{0}^{l}{u}_{x}^{2}\phantom{\rule{0.2em}{0ex}}dx$.

*x*and time

*t*. Equation (6) was studied by many authors such as Dickey [3], Ball Rivera [4], Tucsnak [5], Kouemou Patchen [6], Aassila [7], Oliveira and Lima [8]; Wu and Tsai [9] considered the following beam equation:

*et al.*[11] showed the existence of a global attractor and an inertial manifold of equation (6) with damping ${u}_{t}$. Ma [12] studied the existence and decay rates for the solution of equation (6) with nonlinear boundary conditions

*et al.*[14] established the existence and exponential decay of the Kirchhoff systems with nonlocal boundary condition. Guedda and Labani [15] gave the sufficient condition of the blow-up of the solution to equation (7) with $g({u}_{t})={u}_{t}$ and dynamic boundary condition. As the related problem, we mention the following:

we refer the reader to [16–19].

When $f=0$ and $M(s)=\beta +ks$, 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 [21]. One can find many references on problem (1)-(5) with $M=0$ and $f=0$, for example, Littman and Markus [23], Andrews *et al.* [24], Conrad and Morgul [25], Rao [26].

Dalsen [21, 22] showed the exponential stability of problem (1)-(5) with $m=1$ and $f=0$. Park *et al.* [27] discussed the existence of the solution of the Kirchhoff equation with dynamic boundary conditions and boundary differential inclusion. Doronin and Larkin [28] and Gerbi and Said-Houari [29] were concerned with the wave equation with dynamic boundary conditions. Recently, Autuori and Pucci [30] studied the global nonexistence of solutions of the p-Kirchhoff system with dynamic boundary condition.

In this paper, we use the idea of references [31] 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 [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 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 [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 ${L}^{p}[0,l]$ and the Sobolev space ${H}_{0}^{1}(0,l)$, ${H}^{2}(0,l)$ with their usual scalar products and norms. Especially, ${\parallel \cdot \parallel}_{p}$ denotes the norm of ${L}^{p}[0,l]$ and $\parallel \cdot \parallel $ the norms ${L}^{2}[0,l]$.

We denote $V=\{u|u\in {H}^{2}(0,l),u(0)={u}_{xx}(0)=0\}$.

**Lemma 2.1**

- (1)
*If*$u\in {H}_{0}^{1}$,*then*$|u(l){|}^{2}\le {\parallel {u}_{x}\parallel}^{2},\phantom{\rule{2em}{0ex}}{\parallel u\parallel}^{2}\le {C}_{0}{\parallel {u}_{x}\parallel}^{2};$(9) - (2)
*If*$u\in V$*and*${u}_{xx}(l)+{u}_{x}(l)=0$,*then*${\parallel u\parallel}^{2}\le {C}_{1}({\parallel {u}_{xx}\parallel}^{2}+{u}_{x}^{2}(l,t))\le {C}_{1}({\parallel {u}_{xx}\parallel}^{2}+{u}_{xx}^{2}(l,t)).$(10)

*Proof*Since $u(0)=0$, we have

and the Cauchy inequality, we can get the result of (10) with the help of (9). □

**Lemma 2.2** [34]

*Let*$\varphi (t)$

*be a non*-

*increasing and nonnegative function on*$[0,\mathrm{\infty})$

*such that*

*then*

*where* *C*, *ω* *are positive constants depending on* $\varphi (0)$ *and other known qualities*.

**Lemma 2.3** [33]

*Suppose that a positive*,

*twice*-

*differentiable function*$H(t)$

*satisfies on*$t\ge 0$

*the inequality*

*where* $\beta >0$, *then there is a* ${t}_{1}<{t}_{2}=\frac{H(0)}{\beta {H}^{\prime}(0)}$ *such that* $H(t)\to \mathrm{\infty}$ *as* $t\to {t}_{1}$.

*u*of problem (1)-(5) means that there exists $T>0$ such that

for all $\phi \in C((0,T),V)\cap {C}^{1}(0,T;{L}^{2}(0,l))$.

*u*of problem (1)-(5) by

## 3 Energy decay of the solution

where ${\lambda}_{1}={(p{B}_{1}^{p})}^{-\frac{1}{p-2}}$, ${E}_{1}=(\frac{1}{2}-\frac{1}{p}){\lambda}_{1}^{2}$. Obviously, ${\mathrm{\Sigma}}_{1}\subset {\mathrm{\Sigma}}_{0}$.

Adapting the idea of Vitillaro [35], we have the following lemma.

**Lemma 3.1** *Suppose that* *u* *is the solution of* (1)-(5), ${u}_{0}\in V$, ${u}_{1}\in {L}^{2}$ *and* $(\parallel {u}_{0xx}\parallel ,E(0))\in {\mathrm{\Sigma}}_{1}$, *then* $(\parallel {u}_{xx}(t)\parallel ,E(t))\in {\mathrm{\Sigma}}_{1}$, *for* $t\ge 0$.

**Lemma 3.2**

*Under the condition of Lemma*3.1

*and*$p>2$,

*then*,

*for*$t\ge 0$,

*Proof*By (14) and (18), we have

where $G(\lambda )=\frac{1}{2}{\lambda}^{2}-{B}_{1}^{p}{\lambda}^{p}$. Note that $G(\lambda )$ has the maximum at ${\lambda}_{1}={(p{B}_{1}^{p})}^{-\frac{1}{p-2}}$ and the maximum value $G({\lambda}_{1})={E}_{1}$. We see that $G(\lambda )$ is increasing in $(0,{\lambda}_{1})$, decreasing in $({\lambda}_{1},\mathrm{\infty})$ and $G(\lambda )\to 0$ as $\lambda \to \mathrm{\infty}$. Since $\parallel {u}_{xx}\parallel <{\lambda}_{1}$, $E(0)<{E}_{1}$, then $\parallel {u}_{xx}\parallel <{\lambda}_{1}$ for any $t\ge 0$, so $G(\parallel {u}_{xx}\parallel )\ge 0$.

so (23) holds. □

**Theorem 3.3**

*Let*$p>2$, $(\parallel {u}_{0xx}\parallel ,E(0))\in {\mathrm{\Sigma}}_{1}$,

*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

*u*and integrating over $[0,l]\times [{t}_{1},{t}_{2}]$, by the boundary conditions (2) and (3), we have

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 $E(0)<{E}_{0}$.

**Lemma 4.1**

*Suppose*$p>2$, $(\parallel {u}_{0xx}\parallel ,E(0))\in {\mathrm{\Sigma}}^{e}$,

*then*

*Proof*Since $E(t)$ is non-increasing, and $0<E(0)<{E}_{0}$, then $0<E(t)<{E}_{0}$ for $t\ge 0$. Similar to the proof of Lemma 3.2, we have

Otherwise, we suppose that $\parallel {u}_{xx}({t}_{0})\parallel <{\lambda}_{2}$ for some ${t}_{0}>0$ and by the continuity of $\parallel {u}_{xx}\parallel $, we can choose ${t}_{0}$ such that $\parallel {u}_{xx}({t}_{0})\parallel <{\lambda}_{0}$.

This is impossible since $E(t)\le E(0)$ for all $t\ge 0$. Hence, (45) holds. Furthermore, (43) is established since ${\lambda}_{2}>{\lambda}_{0}$. □

**Theorem 4.2** *Suppose that* *u* *is the local solution of problem* (1)-(5), $p>2(m+1)$, $E(0)<0$, *then the solution* *u* *blows up at some finite time*.

*Proof*Let

*β*are positive constants which will be fixed later (see Levine [33]). Then one finds

Noticing $F(0)>0$, by Lemma 2.3, we get the result. □

**Theorem 4.3**

*Suppose that*$u(x,t)$

*is the local solution of problem*(1)-(5), $p>2(m+1)$,

*and that either of the following conditions is satisfied*:

- (i)
$E(0)=0$

*and*$({u}_{0},{u}_{1})+{u}_{0}(l){u}_{1}(l)>0$; - (ii)
$0<E(0)<{E}_{0}$

*and*$\parallel {u}_{0xx}\parallel >{\lambda}_{0}$ (*or*$(\parallel {u}_{0xx}\parallel ,E(0))\in {\mathrm{\Sigma}}^{e}$);

*then the solution* *u* *blows up at some finite time*.

*Proof*(i) For $E(0)=0$, similar to the proof of Theorem 4.2, we take $\beta =0$ in (51), then (53) holds. Since $F(0)>0$, ${F}^{\prime}(0)=2({u}_{0},{u}_{1})+2{u}_{0}(l){u}_{1}(l)>0$, then the result holds by Lemma 2.3.

- (ii)For the case of $0<E(0)<{E}_{0}$, from (48), (49), (50) and (14), we get$\begin{array}{rl}{F}^{\u2033}(t)=& 2[{\parallel {u}_{t}\parallel}^{2}+{u}_{t}^{2}(l,t)-{\parallel {u}_{xx}\parallel}^{2}-{u}_{xx}^{2}(l,t)-{\parallel {u}_{x}\parallel}^{2}-k{\parallel {u}_{x}\parallel}^{2(m+1)}+\beta ]\\ +p[{\parallel {u}_{t}\parallel}^{2}+{u}_{t}^{2}(l,t)+{\parallel {u}_{xx}\parallel}^{2}+{u}_{xx}^{2}(l,t)+{\parallel {u}_{x}\parallel}^{2}+k{\parallel {u}_{x}\parallel}^{2(m+1)}-2E(t)]\\ =& (p+2)[{\parallel {u}_{t}\parallel}^{2}+{u}_{t}^{2}(l,t)]+(p-2)[{\parallel {u}_{xx}\parallel}^{2}+{u}_{xx}^{2}(l,t)+{\parallel {u}_{x}\parallel}^{2}\\ +(\frac{p}{m+1}-2)k{\parallel {u}_{x}\parallel}^{2(m+1)}]+2\beta -2pE(t).\end{array}$(55)

The remainder of the proof is the same as the proof of Theorem 4.2. □

## Declarations

### 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).

## Authors’ Affiliations

## References

- Woinowsky-Krieger S: The effect of axial force on the vibration of hinged bars.
*J. Appl. Mech.*1980, 17: 35–36.MathSciNetGoogle Scholar - Ball J: Stability theory for an extensible beam.
*J. Differ. Equ.*1973, 14: 58–66.View ArticleGoogle Scholar - Dickey KW: Infinite systems of nonlinear oscillation equations with linear damping.
*SIAM J. Appl. Math.*1970, 7: 208–214.MathSciNetView ArticleGoogle Scholar - 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.1996.0091MathSciNetView ArticleGoogle Scholar - Tucsnak M: Semi-internal stabilization for a nonlinear Euler-Bernoulli equation.
*Math. Methods Appl. Sci.*1996, 9: 897–907.MathSciNetView ArticleGoogle Scholar - Patcheu SK: On a global solution and asymptotic behavior for the generalized damped extensible beam equation.
*J. Differ. Equ.*1997, 135(2):123–138.View ArticleGoogle Scholar - Aassila M: Decay estimate for a quasi-linear wave equation of Kirchhoff type.
*Adv. Math. Sci. Appl.*1999, 9(1):371–381.MathSciNetGoogle Scholar - Oliveira ML, Lima OA: Exponential decay of the solutions of the beams system.
*Nonlinear Anal.*2000, 42: 1271–1291. 10.1016/S0362-546X(99)00155-8MathSciNetView ArticleGoogle Scholar - Wu ST, Tsai LY: Existence and nonexistence of global solutions for a nonlinear wave equation.
*Taiwan. J. Math.*2009, 13B(6):2069–2091.MathSciNetGoogle Scholar - 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.1670150406MathSciNetView ArticleGoogle Scholar - 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.Google Scholar - Ma TF: Boundary stabilization for a nonlinear beam on elastic bearings.
*Math. Methods Appl. Sci.*2001, 24: 583–594. 10.1002/mma.230MathSciNetView ArticleGoogle Scholar - 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.Google Scholar
- 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.MathSciNetView ArticleGoogle Scholar - 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.MathSciNetGoogle Scholar - 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.2007.11.011MathSciNetView ArticleGoogle Scholar - 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.View ArticleGoogle Scholar - 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.2008.09.010MathSciNetView ArticleGoogle Scholar - 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.am1.2003.07.014MathSciNetView ArticleGoogle Scholar - 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-FMathSciNetView ArticleGoogle Scholar - 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/S0218202599000191MathSciNetView ArticleGoogle Scholar - 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/S0218202594000066MathSciNetView ArticleGoogle Scholar - 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/BF01766154MathSciNetView ArticleGoogle Scholar - 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.1996.0053MathSciNetView ArticleGoogle Scholar - 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/S0363012996302366MathSciNetView ArticleGoogle Scholar - Rao BP: Uniform stabilization of a hybrid system of elasticity.
*SIAM J. Control Optim.*1995, 33: 440–454. 10.1137/S0363012992239879MathSciNetView ArticleGoogle Scholar - 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.Google Scholar - 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.MathSciNetView ArticleGoogle Scholar - 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.MathSciNetGoogle Scholar - Autuori G, Pucci P: Kirchhoff system with dynamic boundary conditions.
*Nonlinear Anal.*2010, 73: 1952–1965. 10.1016/j.na.2010.05.024MathSciNetView ArticleGoogle Scholar - 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.1999.6528MathSciNetView ArticleGoogle Scholar - Payne L, Sattinger D: Saddle points and instability on nonlinear hyperbolic equations.
*Isr. J. Math.*1973, 22: 273–303.MathSciNetView ArticleGoogle Scholar - Levine HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.
*SIAM J. Math. Anal.*1974, 5: 138–146. 10.1137/0505015MathSciNetView ArticleGoogle Scholar - 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.MathSciNetGoogle Scholar - 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/S0017089502030045MathSciNetView ArticleGoogle Scholar

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