2. The Global Existence and Nonexistence

doi:10.1155/2010/357404

Research Article

Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation

Yaojun Ye

Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Correspondence should be addressed to Yaojun Ye,yeyaojun2002@yahoo.com.cn Received 14 December 2009; Revised 21 May 2010; Accepted 4 August 2010 Academic Editor: Tocka Diagana

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 hyperbolic equations with strong dissipative termutt−_n

i1∂/∂xi|∂u/∂xi|^p−2∂u/∂xi−aΔutb|u|^r−2uin a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set inW₀^1,pΩand showing the exponential decay of the energy of global solutions through the use of an important lemma of V. Komornik.

1. Introduction

We are concerned with the global solvability and exponential asymptotic stability for the following hyperbolic equation in a bounded domain:

utt−Δpu−aΔutb|u|^r−2u, x∈Ω, t >0 1.1

with initial conditions

ux,0 u0x, utx,0 u1x, x∈Ω 1.2

and boundary condition

ux, t 0, x∈∂Ω, t≥0, 1.3

whereΩis a bounded domain inRⁿ with a smooth boundary∂Ω,a, b >0 andr, p > 2 are real numbers, andΔp_n

i1∂/∂xi |∂/∂xi|^p−2∂/∂xiis a divergence operatordegenerate Laplace operatorwithp >2, which is called ap-Laplace operator.

Equations of type 1.1 are used to describe longitudinal motion in viscoelasticity mechanics and can also be seen as field equations governing the longitudinal motion of a viscoelastic configuration obeying the nonlinear Voight model1–4 .

Forb0, it is well known that the damping term assures global existence and decay of the solution energy for arbitrary initial data4–6 . Fora0, the source term causes finite time blow up of solutions with negative initial energy ifr > p7 .

In8–10 , Yang studied the problem1.1–1.3and obtained global existence results under the growth assumptions on the nonlinear terms and initial data. These global existence results have been improved by Liu and Zhao 11 by using a new method. As for the nonexistence of global solutions, Yang12 obtained the blow up properties for the problem 1.1–1.3with the following restriction on the initial energyE0 < min{−rk1pk2/r − p^1/δ,−1}, wherer > pandk1, k2, andδare some positive constants.

Because thep-Laplace operatorΔpis nonlinear operator, the reasoning of proof and computation are greatly different from the Laplace operatorΔ _n

i1∂²/∂x²_i. By means of the Galerkin method and compactness criteria and a difference inequality introduced by Nakao13 , Ye14,15 has proved the existence and decay estimate of global solutions for the problem1.1–1.3with inhomogeneous termfx, tandp≥r.

In this paper we are going to investigate the global existence for the problem1.1–

1.3 by applying the potential well theory introduced by Sattinger 16 , and we show the exponential asymptotic behavior of global solutions through the use of the lemma of Komornik17 .

We adopt the usual notation and convention. LetW^k,pΩdenote the Sobolev space with the normu_W^k,p_Ω

|α|≤kD^αu^p_LpΩ^1/pandW₀^k,pΩdenote the closure inW^k,pΩ ofC^∞₀ Ω. For simplicity of notation, hereafter we denote by · _pthe Lebesgue spaceL^pΩ norm, · denotes L²Ω norm, and write equivalent norm ∇ · _p instead of W₀^1,pΩ norm · _W^1,p

0 Ω. Moreover,Mdenotes various positive constants depending on the known constants, and it may be different at each appearance.

2. The Global Existence and Nonexistence

In order to state and study our main results, we first define the following functionals:

Ku ∇u^pp−bu^r_r,

Ju 1

p∇u^pp−b ru^r_r,

2.1

foru∈W₀^1,pΩ. Then we define the stable setHby

H

u∈W₀^1,pΩ, Ku>0, Ju< d

∪ {0}, 2.2

where

dinf

sup

λ>0

Jλu, u∈W₀^1,pΩ/{0}

. 2.3

We denote the total energy associated with1.1–1.3by Et 1

2ut² 1

p∇u^pp−b

ru^r_r 1

2ut²Ju 2.4

foru∈W₀^1,pΩ,t≥0, andE0 1/2u1²Ju0is the total energy of the initial data.

Definition 2.1. The solutionux, tis called the weak solution of the problem1.1–1.3on Ω×0, T, ifu∈L^∞0, T;W₀^1,pΩandut∈L^∞0, T;L²Ωsatisfy

ut, v− _t

0

Δpu, v dτa∇u,∇v b _t

0

|u|^r−2u, v

dτ u1, v a∇u0,∇v 2.5

for allv∈W₀^1,pΩandux,0 u0xinW₀^1,pΩ, utx,0 u1xinL²Ω.

We need the following local existence result, which is known as a standard onesee 14,18,19 .

Theorem 2.2. Suppose that 2 < p < r < np/n−pif p < nand 2 < p < r < ∞if n ≤ p. If u0 ∈W₀^1,pΩ, u1 ∈L²Ω, then there existsT >0 such that the problem1.1–1.3has a unique local solutionutin the class

u∈L^∞

0, T;W₀^1,pΩ

, ut∈L^∞

0, T;L²Ω

. 2.6

For latter applications, we list up some lemmas.

Lemma 2.3 see 20, 21 . Let u ∈ W₀^1,pΩ, then u ∈ L^qΩ, and the inequality u_q ≤ Cu_W^1,p

0 Ωholds with a constantC >0 depending onΩ, p, andq, provided that,i2≤q <∞if 2≤n≤pandii2≤q≤np/n−p, 2< p < n.

Lemma 2.4. Letut, xbe a solution to problem1.1–1.3. ThenEtis a nonincreasing function fort >0 and

d

dtEt −a∇utt². 2.7

Proof. By multiplying1.1byutand integrating overΩ, we get 1

2 d

dtut²1 p

d

dt∇u^pp− b r

d

dtu^r_r −a∇utt², 2.8

which implies from2.4that d

dtEut −a∇utt²≤0. 2.9

Therefore,Etis a nonincreasing function ont.

Lemma 2.5. Letu∈W₀^1,pΩ; if the hypotheses inTheorem 2.2hold, thend >0.

Proof. Since

Jλu λ^p

p ∇u^pp− bλ^r

r u^r_r, 2.10

so, we get

d

dλJλu λ^p−1∇u^pp−bλ^r−1u^r_r. 2.11 Letd/dλJλu 0, which implies that

λ1 b^−1/r−p u^r_r

∇u^pp

_−1/r−p

. 2.12

Asλλ1, an elementary calculation shows that d²

dλ²Jλu<0. 2.13

Hence, we have fromLemma 2.3that

sup

λ≥0Jλu Jλ1u r−p

rp b^−p/r−p u_r

∇u_p

_−rp/r−p

≥ r−p

rp bC^r−p/r−p >0.

2.14

We get from the definition ofdthatd >0.

Lemma 2.6. Letu∈H, then

r−p

rp ∇u^pp< Ju. 2.15

Proof. By the definition ofKuandJu, we have the following identity:

rJu Ku r−p

p ∇u^pp. 2.16

Sinceu∈H, so we haveKu>0. Therefore, we obtain from2.16that r−p

rp ∇u^pp≤Ju. 2.17

In order to prove the existence of global solutions for the problem1.1-1.3, we need the following lemma.

Lemma 2.7. Suppose that 2 < p < r < np/n−pifp < n and 2 < p < r < ∞ifn ≤ p. If u0∈H, u1∈L²Ω, andE0< d, thenu∈H, for eacht∈0, T.

Proof. Assume that there exists a numbert^∗∈0, Tsuch thatut∈Hon0, t^∗andut^∗/∈H.

Then, in virtue of the continuity ofut, we see thatut^∗∈∂H. From the definition ofHand the continuity ofJutandKutint, we have either

Jut^∗ d, 2.18

or

Kut^∗ 0. 2.19

It follows from2.4that

Jut^∗ 1

p∇ut^∗pp−b

rut^∗r_r ≤Et^∗≤E0< d. 2.20 So, case2.18is impossible.

Assume that2.19holds, then we get that d

dλJλut^∗ λ^p−1

1−λ^r−p ∇u^pp. 2.21

We obtain fromd/dλJλut^∗ 0 thatλ1.

Since

d²

dλ²Jλut^∗

λ1

−

r−p ∇ut^∗_p<0, 2.22

consequently, we get from2.20that sup

λ≥0Jλut^∗ Jλut^∗|_λ1Jut^∗< d, 2.23 which contradicts the definition ofd. Therefore, case2.19is impossible as well. Thus, we conclude thatut∈Hon0, T.

Theorem 2.8. Assume that 2 < p < r < np/n−pifp < nand 2< p < r < ∞ifn≤ p.ut is a local solution of problem1.1–1.3on0, T. Ifu0 ∈H, u1 ∈L²Ω, andE0< d, then the solutionutis a global solution of the problem1.1–1.3.

Proof. It suffices to show thatut²∇u^ppis bounded independently oft.

Under the hypotheses in Theorem 2.8, we get from Lemma 2.7 that ut ∈ H on 0, T. So formula2.15 inLemma 2.6holds on0, T. Therefore, we have from2.15and Lemma 2.4that

1

2ut²r−p

rp ∇u^pp≤ 1

2ut²Ju Et≤E0< d. 2.24 Hence, we get

ut²∇u^pp≤max

2, rp r−p

d <∞. 2.25

The above inequality and the continuation principle lead to the global existence of the solution, that is,T ∞. Thus, the solutionutis a global solution of the problem1.1–

1.3.

Now we employ the analysis method to discuss the blow-up solutions of the problem 1.1–1.3in finite time. Our result reads as follows.

Theorem 2.9. Suppose that 2 < p < r < np/n−pif p < nand 2 < p < r < ∞if n ≤ p. If u0∈H, u1∈L²Ω, assume that the initial value is such that

E0< Q0, u0_r > S0, 2.26

where

Q0 r−p

rp C^pr/p−r, S0C^p/p−r 2.27

withC >0 is a positive Sobolev constant. Then the solution of the problem1.1–1.3does not exist globally in time.

Proof. On the contrary, under the conditions inTheorem 2.9, letux, tbe a global solution of the problem1.1–1.3; then byLemma 2.3, it is well known that there exists a constantC >0 depending only onn, p, andrsuch thatu_r ≤C∇u_pfor allu∈W₀^1,pΩ.

From the above inequality, we conclude that

∇u^pp≥C^−pu^pr. 2.28

By using2.28, it follows from the definition ofEtthat

Et 1

2ut²Jut 1

2ut²1

p∇u^pp−b ru^r_r

≥ 1

p∇u^pp−b

ru^r_r ≥ 1

pC^pu^pr −b ru^r_r.

2.29

Setting

sst ut_r

Ω|ux, t|^rdx _1/r

, 2.30

we denote the right side of2.29byQs Qut_r, then

Qs 1

pC^ps^p−b

rs^r, s≥0. 2.31

We have

Qs C^−ps^p−1−bs^r−1. 2.32

LettingQt 0, we obtainS0 bC^p1/p−r. AssS0, we have

Qs

sS0 p−1

C^p s^p−2−br−1s^r−2 sS0

p−r b^p−2C^r−2p_1/p−r

<0. 2.33

Consequently, the functionQshas a single maximum valueQ0atS0, where

Q0QS0

1

pC^pbC^pp/p−r−b

rbC^pr/p−r r−p

rp b^pC^pr1/p−r. 2.34

Since the initial data is such thatE0, s0satisfies

E0< Q0, u0_r > S0. 2.35

Therefore, fromLemma 2.4we get

Eut≤E0< Q0, ∀t >0. 2.36

At the same time, by2.29and2.31, it is clear that there can be no timet >0 for which

Eut< Q0, st S0. 2.37

Hence we have alsost> S0for allt >0 from the continuity ofEutandst.

According to the above contradiction, we know that the global solution of the problem 1.1–1.3does not exist, that is, the solution blows up in some finite time.

This completes the proof ofTheorem 2.9.

3. The Exponential Asymptotic Behavior

Lemma 3.1see17 . Letyt:R → Rbe a nonincreasing function, and assume that there is a constantA >0 such that

_∞

s

ytdt≤Ays, 0≤s <∞, 3.1

thenyt≤y0e^1−t/A, for allt≥0.

The following theorem shows the exponential asymptotic behavior of global solutions of problem1.1–1.3.

Theorem 3.2. If the hypotheses inTheorem 2.8are valid, then the global solutions of problem1.1–

1.3have the following exponential asymptotic behavior:

1

2ut² r−p

rp ∇u^pp≤E0e^1−t/M, ∀t≥0. 3.2

Proof. Multiplying byuon both sides of1.1and integrating overΩ×S, T gives

0 _T

S

Ωu

utt−Δpu−aΔut−bu|u|^r−2

dx dt, 3.3

where 0≤S < T <∞.

Since

_T

S

Ωuuttdx dt

Ωuutdx ^T

S

− _T

S

Ω|ut|²dx dt, 3.4

so, substituting the formula3.4into the right-hand side of3.3gives

0 _T

S

Ω

|ut|² 2

p|∇u|^pp−2b r |u|^r

dx dt

− _T

S

Ω

2|ut|²−a∇ut∇u dx dt

Ωuutdx ^T

S

b 2

r −1

T S

u^r_rdtp−2 p

_T

S

∇u^ppdt.

3.5

By exploitingLemma 2.3and2.24, we easily arrive at

but^r_r ≤bC^r∇ut^r_pbC^r∇ut^r−pp ∇ut^pp

< bC^r rpd

r−p

_r−p/p∇ut^pp.

3.6

We obtain from3.6and2.24that

b

1−2 r

u^r_r ≤bC^r rpd

r−p

r−p/p

r−2

r ∇ut^pp

≤bC^r rpd

r−p

_r−p/p r−2

r · rp r−pEt bpr−2C^r

r−p

rpd r−p

_r−p/p Et, p−2

p _T

S

∇u^ppdx dt≤ r p−2 r−p

_T

S

Etdt.

3.7

It follows from3.7and3.5that

2−bpr−2C^r r−p

rpd r−p

_r−p/p

−r p−2 r−p

T S

Etdt

≤ _T

S

Ω

2|ut|²−a∇ut∇u dx dt−

Ωuutdx ^T

S

.

3.8

We have from H ¨older inequality,Lemma 2.3and2.24that

−

Ωuutdx ^T

S

≤

C^prp

r−p ·r−p

rp ∇u^pp1

2ut^{2 T}

S

≤max C^prp

r−p,1

Et|^T_S≤MES.

3.9

Substituting the estimates of3.9into3.8, we conclude that

2−bpr−2C^r r−p

rpd r−p

_r−p/p

−r p−2 r−p

T S

Etdt

≤ _T

S

Ω

2|ut|²−a∇ut∇u

dx dtMES.

3.10

We get fromLemma 2.3andLemma 2.4that

2 _T

S

Ω|ut|²dx dt2 _T

S

ut²dt≤2C² _T

S

∇ut²dt

−2C²

a ET−ES≤ 2C²

a ES.

3.11

From Young inequality, Lemmas2.3and2.4, and2.24, it follows that

−a _T

S

Ω∇u∇utdx dt≤a _T

S

εC²∇u²_pMε∇ut² dt

≤ aC²rpε r−p

_T

S

EtdtMεES−ET

≤ aC²rpε r−p

_T

S

EtdtMεES.

3.12

Choosingεsmall enough, such that 1

2

bpr−2C^r r−p

rpd r−p

_r−p/p r

p−2

r−p aC²rpε r−p

<1, 3.13

and, substituting3.11and3.12into3.10, we get _T

S

Etdt≤MES. 3.14

We letT → ∞in3.14to get

_∞

S

Etdt≤MES. 3.15

Therefore, we have from3.15andLemma 3.1that

Et≤E0e^1−t/M, t∈0,∞. 3.16

We conclude fromu∈H,2.4and3.16that 1

2ut² r−p

rp ∇u^pp≤E0e^1−t/M, ∀t≥0. 3.17

The proof ofTheorem 3.2is thus finished.

Acknowledgments

This paper was supported by the Natural Science Foundation of Zhejiang Province no.

Y6100016, the Science and Research Project of Zhejiang Province Education Commissionno.

Y200803804 and Y200907298. The Research Foundation of Zhejiang University of Science and Technologyno. 200803, and the Middle-aged and Young Leader in Zhejiang University of Science and Technology2008–2010.

References

1 G. Andrews, “On the existence of solutions to the equationutt−uxtxσux_x,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980.

2 G. Andrews and J. M. Ball, “Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity,” Journal of Differential Equations, vol. 44, no. 2, pp. 306–341, 1982.

3 D. D. Ang and P. N. Dinh, “Strong solutions of quasilinear wave equation with non-linear damping,”

SIAM Journal on Mathematical Analysis, vol. 19, pp. 337–347, 1985.

4 S. Kawashima and Y. Shibata, “Global existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992.

5 A. Haraux and E. Zuazua, “Decay estimates for some semilinear damped hyperbolic problems,”

Archive for Rational Mechanics and Analysis, vol. 100, no. 2, pp. 191–206, 1988.

6 M. Kop´aˇckov´a, “Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,”

Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 4, pp. 713–719, 1989.

7 J. M. Ball, “Remarks on blow-up and nonexistence theorems for nonlinear evolution equations,” The Quarterly Journal of Mathematics. Oxford, vol. 28, no. 112, pp. 473–486, 1977.

8 Z. Yang, “Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms,” Mathematical Methods in the Applied Sciences, vol. 25, no. 10, pp. 795–814, 2002.

9 Z. Yang and G. Chen, “Global existence of solutions for quasi-linear wave equations with viscous damping,” Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 604–618, 2003.

10 Y. Zhijian, “Initial boundary value problem for a class of non-linear strongly damped wave equations,” Mathematical Methods in the Applied Sciences, vol. 26, no. 12, pp. 1047–1066, 2003.

11 L. Yacheng and Z. Junsheng, “Multidimensional viscoelasticity equations with nonlinear damping and source terms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 6, pp. 851–865, 2004.

12 Z. Yang, “Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms,” Mathematical Methods in the Applied Sciences, vol. 25, no. 10, pp. 825–833, 2002.

13 M. Nakao, “A difference inequality and its application to nonlinear evolution equations,” Journal of the Mathematical Society of Japan, vol. 30, no. 4, pp. 747–762, 1978.

14 Y. Ye, “Existence of global solutions for some nonlinear hyperbolic equation with a nonlinear dissipative term,” Journal of Zhengzhou University, vol. 29, no. 3, pp. 18–23, 1997.

15 Y. Ye, “On the decay of solutions for some nonlinear dissipative hyperbolic equations,” Acta Mathematicae Applicatae Sinica. English Series, vol. 20, no. 1, pp. 93–100, 2004.

16 D. H. Sattinger, “On global solution of nonlinear hyperbolic equations,” Archive for Rational Mechanics and Analysis, vol. 30, pp. 148–172, 1968.

17 V. Komornik, Exact Controllability and Stabilization, Research in Applied Mathematics, Masson, Paris, France, 1994.

18 Y. Ye, “Existence and nonexistence of global solutions of the initial-boundary value problem for some degenerate hyperbolic equation,” Acta Mathematica Scientia, vol. 25, no. 4, pp. 703–709, 2005.

19 H. Gao and T. F. Ma, “Global solutions for a nonlinear wave equation with thep-Laplacian operator,”

Electronic Journal of Qualitative Theory of Differential Equations, no. 11, pp. 1–13, 1999.

20 R. A. Adams, Sobolev Spaces, vol. 6 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975.

21 O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, vol. 49 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1985.