Volume 2010, Article ID 394859,14pages doi:10.1155/2010/394859
Research Article
Existence and Asymptotic Behavior of Global Solutions for a Class of Nonlinear Higher-Order Wave Equation
Yaojun Ye
Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
Correspondence should be addressed to Yaojun Ye,[email protected] Received 5 November 2009; Accepted 28 January 2010
Academic Editor: Marta Garc´ıa-Huidobro
Copyrightq2010 Yaojun Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The initial boundary value problem for a class of nonlinear higher-order wave equation with damping and source term utt Au a|ut|p−1ut b|u|q−1u in a bounded domain is studied, whereA −Δm,m ≥ 1 is a nature number, anda, b > 0 andp, q > 1 are real numbers. The existence of global solutions for this problem is proved by constructing the stable sets and shows the asymptotic stability of the global solutions as time goes to infinity by applying the multiplier method.
1. Introduction
In this paper we consider the existence and asymptotic behavior of global solutions for the initial boundary problem of the nonlinear higher-order wave equation with nonlinear damping and source term:
uttAua|ut|p−1utb|u|q−1u, x∈Ω, t >0, 1.1 ux,0 u0x, utx,0 u1x, x∈Ω, 1.2 Dαux, t 0, |α| ≤m−1, x∈∂Ω, t≥0, 1.3
where A −Δm, m ≥ 1 is a nature number,a, b > 0 and p, q > 1 are real numbers, Ω is a bounded domain ofRN with smooth boundary∂Ω,Δis the Laplace operator, andα α1, α2, . . . , αN,|α|N
i1|αi|, DαN
i1∂αi/∂xαii, x x1, x2, . . . , xN.
When m 1, the existence and uniqueness, as well as decay estimates, of global solutions and blow up of solutions for the initial boundary value problem and Cauchy problem of1.1have been investigated by many people through various approaches and assumptive conditions 1–8 . Rammaha 9 deals with wave equations that feature two competing forces and analyzes the influence of these forces on the long-time behavior of solutions. Barbu et al.10 study the following initial-boundary value problem:
utt−Δu|u|kjut |u|p−1u, x, t∈Ω×0, T≡QT, ux,0 u0x∈H01Ω, utx,0 u1x∈L2Ω,
u0, x, t∈Γ×0, T,
1.4
whereΩis a bounded domain inRNwith a smooth boundaryΓ,jsis aC1convex, real value function defined onR, andjdenotes the derivative ofj. They prove that every generalized solution to the above problem and additional regularity blows up in finite time, whenever the exponentpis greater than the critical valuekm, and the initial energy is negative.
For the following model of semilinear wave equation with a nonlinear boundary dissipation and nonlinear boundaryinteriorsources,
utt Δufu, x, t∈Ω×0,∞,
∂νuugut hu, x, t∈Γ×0,∞, u0 u0x∈H1Ω, ut0∈u1x∈L2Ω,
1.5
where the operators fu, gut, andhu are Nemytskii operators associated with scalar, continuous functionsfs, gs, andhsdefined for s ∈ R. The function gsis assumed monotone. The paper11,12 proves the existence and uniqueness of both local and global solutions of this system on the finite energy space and derive uniform decay rates of the energy whent → ∞.
Whenm2, Guesmia13 considered the equation
utt Δ2uqxugut 0, x∈Ω, t >0 1.6
with initial boundary value conditions1.2and1.3, wheregis a continuous and increasing function with g0 0, and q : Ω → 0,∞ is a bounded function. He prove a global existence and a regularity result of the problem1.6,1.2, and1.3. Under suitable growth conditions ong, he also established decay results for weak and strong solutions. Precisely, In13 , Guesmia showed that the solution decays exponentially ifg behaves like a linear function, whereas the decay is of a polynomial order otherwise. Results similar to the above system, coupled with a semilinear wave equation, have been established by Guesmia14 . As qxugut in 1.6 is replaced by Δ2ut ΔgΔu. Aassila and Guesmia 15 have obtained a exponential decay theorem through the use of an important lemma of Komornik 16 . Moreover, Messaoudi17 sets up an existence result of this problem and shows that the solution continues to exist globally ifp≥q; however, it blows up in finite time ifp < q.
Nakao18 has used Galerkin method to present the existence and uniqueness of the bounded solutions, and periodic and almost periodic solutions to the problem 1.1–1.3 as the dissipative term is a linear function νut. Nakao and Kuwahara19 studied decay estimates of global solutions to the problem1.1–1.3by using a difference inequality when the dissipative term is a degenerate caseaxut. When there is no dissipative term in1.1, Brenner and von Wahl20 proved the existence and uniqueness of classical solutions to the initial boundary problem for1.1in Hilbert space. Pecher21 investigated the existence and uniqueness of Cauchy problem for1.1by the use of the potential well method due to Payne and Sattinger6 and Sattinger22 .
Whena 0, for the semilinear higher-order wave equation1.1, Wang23 shows that the scattering operators map a band inHsintoHs if the nonlinearities have critical or subcritical powers inHs. Miao24 obtains the scattering theory at low energy using time- space estimates and nonlinear estimates. Meanwhile, he also gives the global existence and uniqueness of solutions under the condition of low energy.
The proof of global existence for problem1.1–1.3is based on the use of the potential well theory 6, 22 . See also Todorova 7, 25 for more recent work. And we study the asymptotic behavior of global solutions by applying the lemma of Komornik16 .
We adopt the usual notation and convention. LetHkΩdenote the Sobolev space with the norm u HkΩ
|α|≤k Dαu 2L2Ω1/2, letH0kΩdenote the closure inHkΩofC∞0 Ω.
For simplicity of notation, hereafter we denote by · r the Lebesgue spaceLrΩnorm and · denotesL2Ωnorm, we write equivalent norm A1/2· instead ofH0mΩnorm · Hm
0Ω. Moreover,Mdenotes various positive constants depending on the known constants and may be different at each appearance.
This paper is organized as follows. In the next section, we will study the existence of global solutions of problem1.1–1.3. Then inSection 3, we are devoted to the proof of decay estimate.
We conclude this introduction by stating a local existence result, which is known as a standard onesee17 .
Theorem 1.1. Suppose thatp, q >1 satisfies
1< q <∞, N≤2m; 1< q≤ N
N−2m, N >2m, 1.7 1< p <∞, N≤2m; 1< p≤ N2m
N−2m, N >2m, 1.8
and u0, u1 ∈ H0mΩ×L2Ω, then there existsT > 0 such that the problem1.1–1.3has a unique local solutionutin the class
u∈C
0, T;H0mΩ
, ut∈C
0, T;L2Ω
∩Lp1Ω×0, T. 1.9
Theorem 1.2. Under the assumptions inTheorem 1.1, if sup
0≤t≤Tmax
utt 2 A1/2ut 2
<∞, 1.10
thenTmax ∞, where0, Tmax is the maximum time interval on which the solution ux, t of problem1.1–1.3exists.
Please notice that in17 , we can also construct the following spaceXT in proving the existence of local solution by using contraction mapping principle:
XT u∈C
0, T ;H0mΩ
, ut∈C
0, T ;L2Ω
, 1.11
which is equipment with norm
ut XT sup
0≤t≤T
1 2
utt 2 A1/2ut 2
. 1.12
Letε >0, and
Xε,T
u∈XT : u XT ≤ε
. 1.13
We define a distancedu, v u−v XT onXε,T, and thenXε,T is a complete distance space.
This show that, for small enough ε, there exists an unique fixed point on Xε,T and T only depends onε. Therefore, with the standard extension method of solution, we obtainTmax
∞for
sup
0≤t≤Tmax
utt 2 A1/2ut 2
<∞. 1.14
Here we omit the detailed proof of extension.
2. The Global Existence
In order to state and prove our main results, we first define the following functionals:
Iu Iut A1/2ut 2−b ut q1q1, Ju Jut 1
2
A1/2ut 2− b
q1 ut q1q1,
2.1
and according to paper18,24 we put
dinf
sup
λ>0
Jλu, u∈H0mΩ/{0}
. 2.2
Then, for the problem1.1–1.3, we are able to define the stable set
W
u∈H0mΩ, Iu>0
∪ {0}. 2.3
We denote the total energy related to1.1by
Eut 1
2 utt 21 2
A1/2ut 2− b
q1 ut q1q1 1
2 utt 2Jut 2.4
foru∈H0mΩ, t≥0, andEu0 1/2 u1 2Ju0is the total energy of the initial data.
Lemma 2.1. Letrbe a number with 2≤r <∞, N≤2mor 2≤r ≤2N/N−2m, N >2m. Then there is a constantCdepending onΩandrsuch that
u r≤C A1/2u , ∀u∈H0mΩ. 2.5 Lemma 2.2. Assume thatu∈H0mΩ; if 1.7holds, then
d q−1 2
q1 1
bCq1∗ 2/q−1 2.6
is a positive constant, where C∗ is the most optimal constant in Lemma 2.1, namely, C∗ sup u q1/ A1/2u .
Proof. Since
Jλu λ2 2
A1/2u 2−bλq1
q1 u q1q1, 2.7
so, we get
d
dλJλu λ A1/2u 2−bλq u q1q1. 2.8 Letd/dλJλu 0, which implies that
λ1b−1/q−1
⎛
⎝ u q1q1 A1/2u 2
⎞
⎠
−1/q−1
. 2.9
Asλλ1, an elementary calculation shows that
d2
dλ2Jλu<0. 2.10
Thus, we have fromLemma 2.1that
sup
λ≥0Jλu Jλ1u q−1 2
q1b−2/q−1
u q1 A1/2u
−2q1/q−1
≥ q−1 2
q1 1
b2/q−1C2q1/q−1>0.
2.11
We get from the definition ofd
d q−1 2
q1 1
bC∗q12/q−1 >0. 2.12
Lemma 2.3. Let ut be a solution of the problem 1.1–1.3. Then Eut is a nonincreasing function fort >0 and
d
dtEut −a utt p1p1. 2.13
Proof. By multiplying1.1byutand integrating overΩ, we get d
dtEut −a utt p1p1≤0. 2.14
Therefore,Eutis a nonincreasing function ont.
Theorem 2.4. Suppose that 1.7 holds. If u0 ∈ W, u1 ∈ L2Ω and the initial energy satisfies Eu0< d, thenu∈W, for eacht∈0, T.
Proof. Assume that there exists a number t∗ ∈ 0, T such that ut ∈ W on 0, t∗ and ut∗/∈W. Then we have
Iut∗ 0, ut∗/0. 2.15
Sinceut∈Won0, t∗, so it holds that
Jut 1
2
A1/2ut 2− b
q1 ut q1q1
≥ 1 2
A1/2ut 2− 1 q1
A1/2ut 2 q−1 2
q1 A1/2ut 2,
2.16
it follows fromIut∗ 0 that Jut∗ 1
2
A1/2ut∗ 2− b
q1 ut∗ q1q1 q−1 2
q1 A1/2ut∗ 2, 2.17
and therefore, we have from2.16and2.17that A1/2ut 2≤ 2
q1
q−1 Jut≤ 2 q1
q−1 Eut≤ 2 q1
q−1 Eu0 2.18
for allt∈0, t∗ .
We obtain fromLemma 2.2andEu0< dthat
Eu0< q−1 2
q1 1
bC∗q12/q−1, 2.19
which implies that
bCq1∗ 2
q1
q−1 Eu0
q−1/2
<1. 2.20
By exploitingLemma 2.1,2.18, and2.20, we easily arrive at
b u q1q1≤bCq1 A1/2u q1 bCq1 A1/2u q−1 A1/2u 2
≤bC∗q1
⎛
⎝ 2
q1
q−1 Eu0
q−1/2⎞
⎠ A1/2u 2< A1/2u 2 2.21
for allt∈0, t∗ . Therefore, we obtain
Iut∗ A1/2ut∗ 2−b ut∗ q1q1>0, 2.22
which contradicts2.15. Thus, we conclude thatut∈Won0, T.
Theorem 2.5. Assume that1.7and 1.8hold, utis a local solution of problem1.1–1.3. If u0 ∈W, u1 ∈L2Ω, andEu0< d, then the solutionutis a global solution of problem1.1–
1.3.
Proof. We obtain from2.18that
d > Eu0≥Eut 1
2 utt 2Jut
≥ 1
2 utt 2 q−1 2
q1 A1/2u 2≥ q−1 2
q1
utt 2 A1/2u 2 2.23
Therefore,
utt 2 A1/2u 2 ≤ 2 q1
q−1 d <∞. 2.24
It follows fromTheorem 1.2thatux, tis the global solution of problem1.1–1.3.
3. Decay Estimate
The following two lemmas play an important role in studying the decay estimate of global solutions for the problem1.1–1.3.
Lemma 3.1see16 . LetF:R → Rbe a nonincreasing function and assume that there are two constantsβ≥1 andA >0 such that
∞
S
Ftβ1/2dt≤AFS, 0≤S <∞, 3.1
thenFt ≤ CF01t−2/β−1, for allt ≥ 0, ifβ > 1, andFt ≤ CF0e−ωt, for allt ≥ 0, if β1, whereCandωare positive constants independent ofF0.
Lemma 3.2. If the hypotheses inTheorem 2.4hold, then
b ut q1q1≤1−θ A1/2ut 2, ∀t∈0,∞, 3.2 where
θ1−bC∗q1 2
q1
q−1 Eu0 q−1/2
>0. 3.3
Moreover, one has
Iut≥θ A1/2ut 2≥ θ
1−θb ut q1q1, ∀t∈0,∞. 3.4 Proof. We get fromLemma 2.1and2.23that
b u q1q1 ≤bCq1 A1/2u q1 bCq1 A1/2u q−1 A1/2u 2
≤bCq1∗ 2
q1
q−1 Eu0 q−1/2
A1/2u 2.
3.5
Let
θ1−bCq1∗ 2
q1
p Eu0
q−1/2
, 3.6
then we have from2.20that 0< θ <1. Thus, it follows that from3.5
b u q1q1≤1−θ A1/2u 2. 3.7
Meanwhile, we conclude from3.7that
Iu A1/2u 2−b u q1q1≥ A1/2u 2−1−θ A1/2u 2θ A1/2u 2≥ θb
1−θ u q1q1. 3.8
This complete the proof ofLemma 3.2.
Theorem 3.3. If the hypotheses inTheorem 2.5are valid, then the global solutions of problem1.1–
1.3have the following asymptotic behavior:
t→lim∞ utt 0, lim
t→∞
A1/2ut 0. 3.9
LetEt Eut. If one can prove that the energy of the global solution satisfies the estimate T
S
Etp1/2dt≤MES 3.10
for all 0≤S < T <∞, thenTheorem 3.3will be proved byLemma 3.1. The proof ofTheorem 3.3is composed of the following propositions.
Proposition 3.4. Suppose thatux, tis the global solutions of1.1–1.3, then one has T
S
ΩEtp−1/2
|ut|2A1/2u2−b|u|q1
dx dt
≤ T
S
ΩEtp−1/2
2|ut|2−a|ut|p−1utu dx dt
p−1 2
T
S
ΩEtp−3/2EtuutdxdtMESp1/2
3.11
for all 0≤S < T <∞.
Proof. Multiplying byEtp−1/2uon both sides of1.1and integrating overΩ×S, T , we obtain that
T
S
ΩEtp−1/2u
uttAua|ut|p−1ut−bu|u|q−1
dx dt0, 3.12
where 0≤S < T <∞.
Since T
S
ΩEtp−1/2uuttdx dt
ΩEtp−1/2uutdx
T
S
− T
S
ΩEtp−1/2|ut|2dx dt−p−1 2
T
S
ΩEtp−3/2Etuutdx dt,
3.13
so, substituting3.13into the left-hand side of3.12, we get that T
S
ΩEtp−1/2
|ut|2A1/2u2−b|u|q1
dx dt
T
S
ΩEtp−1/2
2|ut|2−a|ut|p−1utu dx dt
p−1 2
T
S
ΩEtp−3/2Etuutdx dt−
ΩEtp−1/2uutdx
T
S
.
3.14
Next we observe from2.23that
−
ΩEtp−1/2uutdx T
S
≤Etp−1/2 1
2 u 21
2 ut 2 T
S
≤Etp−1/2 C2
2
A1/2u 21
2 ut 2
T
S
≤Etp−1/2
q1 C2 q−1
q−1 2
q1 A1/2u 21
2 ut 2
T
S
≤max
q1 C2 q−1 ,1
Etp1/2
T
S
≤MESp1/2.
3.15
Therefore we conclude from3.14and3.15that the estimate3.11holds.
Proposition 3.5. If ux, t is the global solutions of the problem 1.1–1.3, then one has the following estimate:
T
S
Etp1/2dt≤M T
S
ΩEtp−1/2
2|ut|2−a|ut|p−1utu
dx dtMESp1/2. 3.16
Proof. It follows fromLemma 3.2and 0< θ <1 that T
S
ΩEtp−1/2
|ut|2A1/2u2−b|u|q1
dx dt
T
S
Etp−1/2
ut 2Iut dt≥
T
S
Etp−1/2
ut 2θ A1/2u 2 dt
≥2θ T
S
Etp−1/2 1
2 ut 21 2
A1/2u 2 dt≥2θ
T
S
Etp1/2dt.
3.17
We have fromLemma 2.1and2.23that
p−1
2 T
S
ΩEtp−3/2Etuutdx dt
≤ p−1 2
T
S
Etp−3/2Et 1
2 u 21 2 ut 2
dt
≤ −p−1 2
T
S
Etp−3/2Et C2
2
A1/2u 21 2 ut 2
dt
≤ −p−1 2
T
S
Etp−3/2Et
q1 C2 q−1
q−1 2
q1 A1/2u 21 2 ut 2
dt
≤ −p−1 2 max
q1 C2 q−1 ,1
T
S
Etp−1/2Etdt
−p−1 p1max
q1 C2 q−1 ,1
Etp1/2
T
S
≤MESp1/2.
3.18
We get from3.11,3.17, and3.18that
2θ T
S
Etp1/2dt≤ T
S
ΩEtp−1/2
2|ut|2−a|ut|p−1utu
dx dtMESp1/2. 3.19
Therefore we conclude the estimate3.16from3.19.
Proposition 3.6. Letux, tbe the global solutions of the initial boundary problem1.1–1.3, then the following estimate holds:
T
S
Etp1/2dt≤M1E0p−1/2ES. 3.20
Proof. We get from Young inequality and2.13that
2 T
S
ΩEtp−1/2|ut|2dx dt≤ T
S
Ω
ε1Etp1/2Mε1|ut|p1 dx dt
≤Mε1
T
S
Etp1/2dtMε1 T
S
ut p1 p1dt
Mε1 T
S
Etp1/2dt− Mε1
a ET−ES
≤Mε1
T
S
Etp1/2dtMES.
3.21
We receive from Young inequality,Lemma 2.1,2.13, and2.23that
−a T
S
ΩEtp−1/2uut|ut|p−1dxdt
≤a T
S
Etp−1/2
ε2 u p1p1Mε2 ut p1 p1
dt
≤aCp1ε2E0p−1/2 T
S
A1/2u p1dtaMε2ESp−1/2 T
S
ut p1 p1dt
aCp1ε2E0p−1/2 T
S
2 q1 q−1 Et
p1/2
dtMε2ESp−1/2ES−ET
≤aCp1ε2E0p−1/2 2
q1 q−1
p1/2T
S
Etp1/2dtMε2ESp1/2,
3.22
whereMε1andMε2are positive constants depending onε1andε2. ε1andε2are small enough such that
Mε1aE0p−1/2 2
q1 q−1 C2
p1/2
ε2<1, 3.23
and then, substituting3.21and3.22into3.16, we get T
S
Etp1/2dt≤MES MESp1/2≤M1E0p−1/2ES. 3.24
Therefore, we have fromLemma 3.1andProposition 3.6that
Et≤ME01t−p−1/2, t∈0,∞. 3.25
HereME0>0 is a constant depending onE0.
It follows from2.23and3.25that
t→lim∞ utt lim
t→∞
A1/2ut 0. 3.26
The proof ofTheorem 3.3is thus finished.
Acknowledgments
This Research was supported by the Natural Science Foundation of Henan Province no.
200711013, The Science and Research Project of Zhejiang Province Education Commission no. Y200803804, The Research Foundation of Zhejiang University of Science and Technol- ogyno. 200803, and the Middle-aged and Young Leader in Zhejiang University of Science and Technology2008–2012.
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