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Energy decay and blowup of solution for a Kirchhoff equation with dynamic boundary condition
Boundary Value Problemsvolume 2013, Article number: 166 (2013)
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 $f(s)=s{}^{p2}s$, $M(s)=1+{s}^{m}$, $p>2$, $m\ge 1$ are positive constants and ${\parallel {u}_{x}\parallel}^{2}={\int}_{0}^{l}{u}_{x}^{2}\phantom{\rule{0.2em}{0ex}}dx$.
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 $u(x,t)$ is the lateral displacement at the space coordinate 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:
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 ${u}_{t}$. Ma [12] studied the existence and decay rates for the solution of equation (6) with nonlinear boundary conditions
Pazoto and Menzala [13] were concerned with equation (6) with rotational inertia term ${u}_{xxtt}$ and nonlinear boundary condition (8). Santos 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 blowup 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 SaidHouari [29] were concerned with the wave equation with dynamic boundary conditions. Recently, Autuori and Pucci [30] studied the global nonexistence of solutions of the pKirchhoff system with dynamic boundary condition.
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 ${L}^{p}[0,l]$ 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 $V=\{uu\in {H}^{2}(0,l),u(0)={u}_{xx}(0)=0\}$.
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
Take $x=l$, we get the first inequality of (9). Integrating the above inequality over $[0,l]$, 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 [34]
Let $\varphi (t)$ be a nonincreasing 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, twicedifferentiable 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 $t\to {t}_{1}$.
A solution u of problem (1)(5) means that there exists $T>0$ such that
and it satisfies the following identity
for all $\phi \in C((0,T),V)\cap {C}^{1}(0,T;{L}^{2}(0,l))$.
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, $E(t)$ is a nonincreasing function on $[0,\mathrm{\infty})$. Moreover, we have
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 ${\lambda}_{1}={(p{B}_{1}^{p})}^{\frac{1}{p2}}$, ${E}_{1}=(\frac{1}{2}\frac{1}{p}){\lambda}_{1}^{2}$. 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 ${\lambda}_{1}={(p{B}_{1}^{p})}^{\frac{1}{p2}}$ 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 $G(\parallel {u}_{xx}\parallel )\ge 0$.
By (24), we have
then (21) holds since $G(\parallel {u}_{xx}\parallel )>0$. Furthermore, we have
So (22) holds. Similar to (25), the above equality becomes
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
Now, for the above estimate and the mean value theorem, we choose ${t}_{1}\in [t,t+\frac{1}{4}]$ and ${t}_{2}\in [t+\frac{3}{4},t+1]$ such that, for $i=1,2$,
Multiplying equation (1) by u and integrating over $[0,l]\times [{t}_{1},{t}_{2}]$, by the boundary conditions (2) and (3), we have
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 $E(t)$ is nonincreasing, we arrive at
By the Young inequality, (9), (10), (23), (26) and the fact that $E(t)$ is nonincreasing, 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 $E(t)$ is nonincreasing, we choose ${t}_{3}\in [{t}_{1},{t}_{2}]$ such that
Then, using (26), ${t}_{3}<t+1$, and the fact that $E(t)$ is nonincreasing, we have
Since $E(t)$ is nonincreasing, combining this with (40), (38) and (26), we have
Choosing ${\epsilon}_{1}$ sufficiently small, (41) leads to
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 $E(0)<{E}_{0}$.
Lemma 4.1 Suppose $p>2$, $(\parallel {u}_{0xx}\parallel ,E(0))\in {\mathrm{\Sigma}}^{e}$, then
Proof Since $E(t)$ is nonincreasing, 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
where $g(\lambda )=\frac{1}{2}{\lambda}^{2}\frac{1}{p}{B}_{1}^{p}{\lambda}^{p}$. Note that $g(\lambda )$ has the maximum at ${\lambda}_{0}={B}_{1}^{\frac{2p}{p2}}$ and the maximum value is $g({\lambda}_{0})={E}_{0}$. It is easy to verify that $g(\lambda )$ is increasing for $0<\lambda <{\lambda}_{0}$, decreasing in $\lambda >{\lambda}_{0}$ and $g(\lambda )\to \mathrm{\infty}$ as $\lambda \to +\mathrm{\infty}$. Therefore, since $E(0)<{E}_{0}$, there exists ${\lambda}_{2}>{\lambda}_{0}$ such that $g({\lambda}_{2})>E(0)$. By (44), we have $g(\parallel {u}_{0xx}\parallel )\le E(0)=g({\lambda}_{2})$, which implies $\parallel {u}_{0xx}\parallel \ge {\lambda}_{2}$. We claim that
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 $\parallel {u}_{xx}({t}_{0})\parallel <{\lambda}_{0}$.
Again the use of (44) leads to
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
where ${t}_{0}$, ${T}_{0}$, β are positive constants which will be fixed later (see Levine [33]). Then one finds
By (17) and (14), we have
Taking $0<\beta <2E(0)$ and noticing $p>2$, we get
By the Hölder inequality, we have
Denote
Then, by (48), (51), (52) and the CauchySchwarz inequality, we arrive at
Take ${t}_{0}$ sufficiently large such that
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
$$\begin{array}{rl}{F}^{\u2033}(t)=& 2[{\parallel {u}_{t}\parallel}^{2}+{u}_{t}^{2}(l,t){\parallel {u}_{xx}\parallel}^{2}{u}_{xx}^{2}(l,t){\parallel {u}_{x}\parallel}^{2}k{\parallel {u}_{x}\parallel}^{2(m+1)}+\beta ]\\ +p[{\parallel {u}_{t}\parallel}^{2}+{u}_{t}^{2}(l,t)+{\parallel {u}_{xx}\parallel}^{2}+{u}_{xx}^{2}(l,t)+{\parallel {u}_{x}\parallel}^{2}+k{\parallel {u}_{x}\parallel}^{2(m+1)}2E(t)]\\ =& (p+2)[{\parallel {u}_{t}\parallel}^{2}+{u}_{t}^{2}(l,t)]+(p2)[{\parallel {u}_{xx}\parallel}^{2}+{u}_{xx}^{2}(l,t)+{\parallel {u}_{x}\parallel}^{2}\\ +(\frac{p}{m+1}2)k{\parallel {u}_{x}\parallel}^{2(m+1)}]+2\beta 2pE(t).\end{array}$$(55)
By Lemma 4.1,
Combining (55) with (56), $E(0)<{E}_{0}$ and (17), we get
Take $\beta =2({E}_{0}E(0))>0$, then $2p({E}_{0}E(0))+2\beta =(p+2)\beta >0$, since $p>2$ and $2p>p+2$, then (57) can be rewritten
The remainder of the proof is the same as the proof of Theorem 4.2. □
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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).
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Keywords
 Energy Decay
 Nonlinear Boundary Condition
 Nonlocal Boundary Condition
 Dynamic Boundary Condition
 Inertial Manifold