Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

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R E S E A R C H

Open Access

Existence of weak solutions for viscoelastic

hyperbolic equations with variable exponents

Yunzhu Gao

1*

_{and Wenjie Gao}

2

*_{Correspondence:} [email protected]

1_{Department of Mathematics and} Statistics, Beihua University, Jilin, P.R. China

Full list of author information is available at the end of the article

Abstract

The authors of this paper study a nonlinear viscoelastic equation with variable exponents. By using the Faedo-Galerkin method and embedding theory, the existence of weak solutions is given to the initial and boundary value problem under suitable assumptions.

Keywords: existence; weak solutions; viscoelastic; variable exponents

1 Introduction

Let⊂RN (N≥) be a bounded Lipschitz domain and  <T<∞. Consider the follow-ing nonlinear viscoelastic hyperbolic problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

utt–u–utt+ t

g(t–τ)u(τ)dτ+|ut|m(x)–ut=|u|p(x)–u, (x,t)∈QT,

u(x,t) = , (x,t)∈ST,

u(x, ) =u(x), ut(x, ) =u(x), x∈,

(.)

whereQT=×(,T],STdenotes the lateral boundary of the cylinderQT.

It will be assumed throughout the paper that the exponentsm(x),p(x) are continuous inwith logarithmic module of continuity:

 <m–=inf

x∈m(x)≤m(x)≤m +₌_sup

x∈

m(x) <∞, (.)

 <p–=inf

x∈p(x)≤p(x)≤p +₌_sup

x∈

p(x) <∞, (.)

∀z,ξ∈,|z–ξ|< , m(z) –m(ξ)+p(z) –p(ξ)≤ω|z–ξ| , (.)

where

lim sup τ→+

ω(τ)ln

τ =C< +∞.

And we also assume that

(H) g:R+→R+isCfunction and satisﬁes

g() > ,  – ∞



g(s)ds=l> ;

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(H) there existsη> such that

g(t) < –ηg(t), t≥.

In the case whenm,pare constants, there have been many results about the existence and blow-up properties of the solutions, we refer the readers to the bibliography given in [–].

In recent years, much attention has been paid to the study of mathematical models of electro-rheological ﬂuids. These models include hyperbolic, parabolic or elliptic equations which are nonlinear with respect to gradient of the thought solution and with variable exponents of nonlinearity; see [–] and references therein. Besides, another important application is the image processing where the anisotropy and nonlinearity of the diﬀusion operator and convection terms are used to underline the borders of the distorted image and to eliminate the noise [, ].

To the best of our knowledge, there are only a few works about viscoelastic hyperbolic equations with variable exponents of nonlinearity. In [] the authors studied the ﬁnite time blow-up of solutions for viscoelastic hyperbolic equations, and in [] the authors discussed only the viscoelastic hyperbolic problem with constant exponents. Motivated by the works of [, ], we shall study the existence and energy decay of the solutions to Problem (.) and state some properties to the solutions.

The outline of this paper is the following. In Section , we introduce the function spaces of Orlicz-Sobolev type, give the deﬁnition of the weak solution to the problem and prove the existence of weak solutions for Problem (.) with Galerkin’s method.

2 Existence of weak solutions

In this section, the existence of weak solutions is studied. Firstly, we introduce some Ba-nach spaces

Lp(x)() =

u(x) :uis measurable in,Ap(·)(u) =

|u|p(x)dx<∞

,

up(·)=inf

λ> ,Ap(·)(u/λ)≤.

Lemma .[] For u∈Lp(x)(),the following relations hold:

() up(·)<  (= ; > )⇔Ap(·)(u) <  (= ; > );

() up(·)< ⇒ up

+

p(·)≤Ap(·)(u)≤ up

–

p(·);up(·)> ⇒ up

+

p(·)≥Ap(·)(u)≥ up

–

p(·);

() up(·)→⇔Ap(·)(u)→;up(·)→ ∞ ⇔Ap(·)(u)→ ∞.

Lemma .[, ] For u∈W_,p(·)(),if p satisﬁes condition(.),the p(·)-Poincaré in-equality

up(x)≤C∇up(x)

holds,where the positive constant C depends on p and.

Remark . Note that the following inequality

|u|p(x)dx≤C

|∇u|p(x)dx

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Lemma .[] Letbe an open domain(that may be unbounded)inRN _{with cone} property.If p(x) :→Ris a Lipschitz continuous function satisfying <p–≤p+<N_k and r(x) :→Ris measurable and satisﬁes

p(x)≤r(x)≤p∗(x) = Np(x)

N–kp(x) a.e. x∈,

then there is a continuous embedding Wk,p(x)()→Lr(x)().

The main theorem in this section is the following.

Theorem . Let u,u∈H(),the exponents m(x),p(x)satisfy conditions(.)-(.). Then Problem(.)has at least one weak solution u:×(,∞)→Rin the class

u∈L∞,∞;H_() , u ∈L∞,∞;H_() , u ∈L,∞;H_() .

And one of the following conditions holds:

(A)  <p–_<_p+_<_max_{_N, Np–

N–p–}, <m–<m+<p–; (A) max{,_N+N }<p–_<_p+_{< }_,_{ <}_m–_<_m+_<p–_–

p– < .

Proof Let{wj}∞_j=be an orthogonal basis ofH_() withwj

–wj=λjwj, x∈, wj= , x∈∂.

Vk=span{wi, . . . ,wk}is the subspace generated by the ﬁrstkvectors of the basis{wj}∞_j=. By normalization, we havewj= . Let us deﬁne the operator

Lu,=

utt+∇u∇– t



g(t–τ)∇u∇dτ+|ut|m(x)–ut

–α|u|p(x)–u

dx, ∈Vk.

For any given integerk, we consider the approximate solution

uk= k

i= ck_i(t)wi,

which satisﬁes ⎧ ⎨ ⎩

Luk,wi= , i= , , . . .k,

uk() =uk, ukt() =uk, (.)

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Problem (.) generates the system ofkordinary diﬀerential equations ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (ck

i(t)) = –λicki(t) +λi t

g(t–τ)cki(τ)dτ

+|(k_i=(ck_i(t)),wi)|m(x)–(k_i=(ck_i(t)),wi)

–α|(k_i=ck

i(t),wi)|p(x)–( k

i=cki(t),wi), ck_i() = (u,wi), (ck_i()) = (u,wi), i= , , . . . ,k.

(.)

By the standard theory of the ODE system, we infer that problem (.) admits a unique solutionck

i(t) in [,tk], wheretk> . Then we can obtain an approximate solutionuk(t) for (.), inVk, over [,tk). And the solution can be extended to [,T], for any givenT> , by the estimate below. Multiplying (.) (ck

i(t)) and summing with respect toi, we conclude that

d dt

 uk

 +

 ∇uk

 –

t



g(t–τ)

∇uk(τ)∇uk(t) dx dτ

+

u_km(x)dx

–α d

dt

 p(x)|uk|

p(x)_dx

= . (.)

By simple calculation, we have

– t



g(t–τ)

∇uk(τ),∇u_k(t) dx dτ

= 

d

dt(g ∇uk)(t) –  

g ∇uk (t) –   d dt t 

g(s)ds∇uk

+

g(t)∇uk 

, (.)

here

(ϕ ∇ψ)(t) t



ϕ(t–τ)∇ψ(t) –∇ψ(τ)_dτ.

Combining (.)-(.) and (H)-(H), we get

d dt

 uk

 +

 ∇uk

  +   – t  g(s)ds

∇uk_+

(g ∇uk)(t) –α

 p(x)|uk|

p(x)_dx

= 

g ∇uk (t) – 

g(t)∇uk  –

u_km(x)dx. (.)

Integrating (.) over (,t), and using assumptions (.)-(.), we have

 uk

 +

 ∇uk

 +    – t  g(s)ds

∇uk_

+

(g ∇uk)(t) –α  p(x)|uk|

p(x)_≤_C,

whereC is a positive constant depending only onu_H

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Hence, by Lemma ., we also have

 uk

 +

 ∇uk

 +    – t  g(s)ds

∇uk_+

(g ∇uk)(t)

–max

α 

p–uk p– p(x),α

 p–uk

p+ p(x)

≤C. (.)

In view of (H)-(H) and (A)-(A), we also have

u_k_+∇u_k_+∇uk_+ (g ∇uk)(t)≤C, (.)

whereC is a positive constant depending only onu_H

,uH, l,p

–_,_p+_{. It follows} from (.) that

uk is uniformly bounded inL∞

,T;H_(), (.) u_k is uniformly bounded inL∞,T;H_(). (.) Next, multiplying (.) by (ck_i(t)) and then summing with respect toi, we get that the following holds:

u _k_dx+∇u _k_+ d dt

 m(x)uk

m(x)

= –

∇uk∇u _kdx+ t



g(t–τ)

∇uk(τ)∇u _k(t)dx dτ

+α

|uk|p(x)–uku _kdx. (.)

Note that –

∇uk∇u _kdx≤ε∇u _k_+  ε∇uk



, ε> , (.)

_tg(t–τ)

∇uk(τ)∇u _k(t)dx dτ

≤ε∇u _k_+  ε

t



g(t–τ)∇uk(τ)dτ

 dx

≤ε∇u _k_+  ε t  g(s)ds t 

g(t–τ)

_∇uk(τ) 

dx dτ

≤ε∇u _k_+( –l)g() ε

t



_∇uk(τ)_dτ, (.)

α|uk|p(x)–uku k≤αεu k  +

α

ε|uk|

p(x)–_u k

 

≤αεu _k_+ α ε

|uk|p(x)–uk 

dx.

From Lemma ., we have

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|uk|p(x)–uk dx=

|uk|(p(x)–)dx

≤max

|uk|(p––)dx,

|uk|(p+–)dx

≤maxC∗(p––) ∇_uk

 (p––)_,_C∗



(p+–)∇_uk



(p+–)_, _(.)

whereC,C∗are embedding constants. From (.)-(.), we obtain that

u _kdx+ ( – ε–αεC)∇u _k_+ d dt

 m(x)uk

m(x)

≤ 

ε∇uk

 +

( –l)g() ε

t



_∇uk(τ)  dτ +maxC∗(p––) ∇_u

k



p––_,_C∗

 (p+–)∇_u

k



p+–_. _(.)

Integrating (.) over (,t) and using (.), Lemma ., we get t



u _kdτ+ ( – ε–αεC) t



_∇u _k_dτ+

 m(x)uk

m(x) dx

≤ 

ε

C + ( –l)g()T +C, (.)

whereC is a positive constant depending only onu_H .

Takingα,εsmall enough in (.), we obtain the estimate t



u _kdτ+

 m(x)uk

m(x)

dx≤C.

Hence, by Lemma ., we have t



u _kdτ+min

 m+uk

m–

m(x),  m+uk

m+

m(x)

≤C, (.)

whereC is a positive constant depending only onuH

,uH,l,g(),T.

From estimate (.), we get

u _k is uniformly bounded inL,T;H_(). (.) By (.)-(.) and (.), we infer that there exist a subsequence{ui}of{uk}and a function usuch that

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represented by the same notation, such that

u_i→u strongly inL,T;L() ,

which impliesu_i→u almost everywhere in×(,T). Hence, by (.)-(.),

|ui|p(x)–ui|u|p(x)–u weakly in×(,T), (.) u_im(x)–u_i→um(x)–u almost everywhere in×(,T). (.)

Multiplying (.) byφ(t)∈C(,T) (whichC(,T) is the space ofC∞function with compact support in (,T)) and integrating the obtained result over (,T), we obtain that

Luk,wiφ(t)

= , i= , , . . . ,k. (.)

Note that{wi}∞_i=is a basis ofH(). Convergence (.)-(.) is suﬃcient to pass to the limit in (.) in order to get

utt–u–utt+ t



g(t–τ)u(τ)dτ+|ut|m(x)–ut=|u|p(x)–u, inL,T;H–()

for arbitraryT> . In view of (.)-(.) and Lemma .. in [], we obtain

uk()u() weakly inH_(), u_k()u() weakly inH_().

Hence, we getu() =u,u() =u. Then, the existence of weak solutions is established.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YG carried out the study of existence of weak solutions for viscoelastic hyperbolic equations with variable exponents and drafted the manuscript. WG participated in the discussion of existence of weak solutions for viscoelastic hyperbolic equations with variable exponents.

Author details

1_{Department of Mathematics and Statistics, Beihua University, Jilin, P.R. China.}2_{Institute of Mathematics, Jilin University,} Changchun, P.R. China.

Acknowledgements

Supported by NSFC (11271154) and by Department of Education for Jilin Province (2013439).

Received: 23 June 2013 Accepted: 21 August 2013 Published: 11 September 2013 References

1. Cavalcanti, MM, Domingos Cavalcanti, VN, Ferreira, J: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci.24, 1043-1053 (2001)

2. Cavalcanti, MM, Domingos Cavalcanti, VN, Soriano, JA: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Diﬀer. Equ.2002(44), 1-14 (2002)

3. Cavalcanti, MM, Oquendo, HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim.42(4), 1310-1324 (2003)

4. Messaoudi, SA: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl.320, 902-915 (2006)

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6. Messaoudi, SA, Said-Houari, B: Blow up of solutions of a class of wave equations with nonlinear damping and source terms. Math. Methods Appl. Sci.27, 1687-1696 (2004)

7. Antontsev, SN, Zhikov, V: Higher integrability for parabolic equations ofp(x,t)-Laplacian type. Adv. Diﬀer. Equ.10, 1053-1080 (2005)

8. Lian, SZ, Gao, WJ, Cao, CL, Yuan, HJ: Study of the solutions to a model porous medium equation with variable exponents of nonlinearity. J. Math. Anal. Appl.342, 27-38 (2008)

9. Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functions in image restoration. SIAM J. Appl. Math.66, 1383-1406 (2006)

10. Gao, Y, Guo, B, Gao, W: Weak solutions for a high-order pseudo-parabolic equation with variable exponents. Appl. Anal. (2013). doi:10.1080/00036811.2013.772138

11. Aboulaicha, R, Meskinea, D, Souissia, A: New diﬀusion models in image processing. Comput. Math. Appl.56, 874-882 (2008)

12. Andreu-Vaillo, F, Caselles, V, Mazón, JM: Parabolic Quasilinear Equations Minimizing Linear Growth Functions. Progress in Mathematics, vol. 223. Birkhäuser, Basel (2004)

13. Antontsev, SN: Wave equation withp(x,t)-Laplacian and damping term: blow-up of solutions. C. R., Méc.339, 751-755 (2011)

14. Fan, X, Zhao, D: On the spacesLp(x)_{() and}_Lm,p(x)_{(). J. Math. Anal. Appl.}_{263, 424-446 (2001)} 15. Kováˇcik, O, Rákosník, J: On spacesLp(x)_and_W1,p(x)_{. Czechoslov. Math. J.}_{41(116), 592-618 (1991)}

16. Zhao, JN: Existence and nonexistence of solutions forut= div(|∇u|p–2∇u) +f(∇u,u,x,t). J. Math. Anal. Appl.172,

130-146 (1993)

17. Fan, X, Shen, J, Zhao, D: Sobolev embedding theorems for spacesWk,p(x)_{(). J. Math. Anal. Appl.}_{262, 749-760 (2001)} 18. Lions, JL: Quelques Metodes De Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969) 19. Zheng, SM: Nonlinear Evolution Equation. CRC Press, Boca Raton (2004)

doi:10.1186/1687-2770-2013-208