MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/9349625 9349625 Research Article Strong Solutions and Global Attractors for Kirchhoff Type Equation http://orcid.org/0000-0002-2665-9204 Chen Xiangping 1 Xie Xue-Jun Department of Mathematics Jining University Qufu Shandong 273155 China jnxy.edu.cn 2018 872018 2018 11 05 2018 23 06 2018 872018 2018 Copyright © 2018 Xiangping Chen. 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.

We study the long-time behavior of the Kirchhoff type equation with linear damping. We prove the existence of strong solution and the semigroup associated with the solution possesses a global attractor in the higher phase space.

National Natural Science Foundation of China 61703181
1. Introduction

We consider the following nonlinear Kirchhoff type equation with the initial-boundary conditions:(1)utt+ut+2u+φu=fx,u0=u0,ut0=u1,uΩ=uΩ=0,where Ω is a bounded domain in Rn with smooth boundary, denotes the Laplace operator, f is a given function lying in L2(Ω), independent of time, and φC1(R) with φ(0)=0 fulfills the dissipation inequality(2)liminfuφuu>-λ12,and(3)φ-l,where l>0 is a real number and λ1 is the first eigenvalue of - on L2(Ω) with Dirichlet boundary conditions(4)-w=λw,wΩ=0.

When n=1, this problem describes, for instance, the motion of a vibrating string with fixed boundary in a viscous medium. In particular, the function represents the displacement from equilibrium, ut is the velocity, and the term φ(u)-f may correspond to a (nonlinear) elastic force. For more details on the model of Kirchhoff, one can refer to  and the reference therein.

When the coefficient of ut is a positive function g(u), which depends on u, then the term g(u)ut is a resistance force and the model impresses that the viscous medium embedding the string is stratified. In this case, the existence of the global and exponential attractors has been proven by S. Kolbasin in . The attractor is in the phase space H02(Ω)×L2(Ω) and it is bounded in [H4(Ω)H02(Ω)]×H02(Ω), where Ω is a bounded domain in R3 with smooth boundary.

Similar models have been considered in , such as plate equation(5)utt+σuut+2u+λu+fu=gx,when σ(u)=α(x), the existence, regularity, and finite dimensionality of a global attractor in H22(Rn)×L2(Rn) with a localized damping and a critical exponent were proven in . For Ω a three-dimensional unbounded domain and under suitable conditions on σ(u) and f(u), A.Kh. Khanmamedov in  showed that this equation possesses a global attractor in H2(R3)×L2(R3). About Kirchhoff models, long-time dynamics properties were studied by Yang and I. Chueshov in  and their references. For example, Yang et al. discussed the long-time behavior of solutions to the Cauchy problem of some Kirchhoff type equations with a strong dissipation in [7, 8] and proved that the dynamical system possesses a global attractor under suitable conditions in the phase space X1+δ, where X1+δ=V1+δ×Vδ,0<δ1.

Thus, to the best of our knowledge, the research about global attractors of the weak solutions for problem (1) with respect to the norm of H2×L2 is much more; however, the results about the existence of the strong solutions and strong global attractors for (1) are relatively fewer. The purpose of this paper is to supplement some conclusions for the above problem. In particular, we do not demand the function φ(u) to be bounded by polynomials and present here a method different from . Furthermore, the global attractor is established in the higher energy space in H4(Ω)H02(Ω).

The paper is organized as follows. In Section 2, we recall some notations and some general facts about the dynamical systems theory. In Section 3, we prove well-posedness and the existence of a bounded absorbing set for problem (1). Sections 4 and 5 contain our main results, and we prove the existence of strong solution and a global attractor in the space of higher order.

2. Preliminaries

Throughout the paper we will denote(6)H=L2Ω,V=H01Ω,V1=H2ΩH01Ω,V2=DA=uH2ΩAuL2Ω,whereA=2.

The norm and the scalar product in L2(Ω) is denoted by · and (,), respectively, the norm in Vi is denoted by ·Vi,(i=0,1,2), and the Hilbert spaces E1, E2 are(7)E1=V1×H,E2=V2×V1.

For any given function u(t), we will write for short ξ(t)=(u(t),ut(t)) and endow space E1,E2 with the standard inner product and norm ξuE12=uV12+utL22, ξuE22=uV22+utV12.

For convenience, the letters C and Q present different positive constants and different positive increasing functions, respectively.

We collect some basic concepts and general theorems, which are important for getting our main results. We refer to  and the references therein for more details.

Definition 1 (see [<xref ref-type="bibr" rid="B17">16</xref>, <xref ref-type="bibr" rid="B19">18</xref>]).

Let X be a Banach space and {S(t)}t0 be a family operator on X. We say that {S(t)}t0 is a norm-to-weak continuous semigroup on X, if {S(t)}t0 satisfies

S(0)=Id (the identity);

S(t)S(s)=S(t+s), t,s0;

S(tn)xnS(t)x, if tnt,xnx in X.

Definition 2 (see [<xref ref-type="bibr" rid="B16">15</xref>]).

A C0 semigroup {S(t)}t0 in a Banach space X is said to satisfy the condition (C) if, for any ε>0 and for any bounded set B of X, there exists t(B)>0 and a finite dimensional subspace X1 of X, such that {PS(t)xX,xB,tt(B)} is bounded and(8)I-PStxX<ε,ttB,xB,where P:XX1 is a bounded projector.

Theorem 3 (see [<xref ref-type="bibr" rid="B15">14</xref>]).

Let X be a Banach space and {S(t)}t0 be a norm-to-weak continuous semigroup on X. Then {S(t)}t0 has a global attractor in X provided that the following conditions hold true:

(i) {S(t)}t0 has a bounded absorbing set B0 in X.

(ii) {S(t)}t0 satisfies the condition (C).

Theorem 4 (see [<xref ref-type="bibr" rid="B17">16</xref>, <xref ref-type="bibr" rid="B19">18</xref>]).

Let X and Y be two Banach spaces such that XY with a continuous injection. If a function φ belongs to L(0,T;X) and is weakly continuous with values in Y, then φ is weakly continuous with values in X.

Theorem 5 (see [<xref ref-type="bibr" rid="B17">16</xref>, <xref ref-type="bibr" rid="B19">18</xref>]).

Let X, Y be two Banach spaces and X,Y be their dual spaces, respectively, such that(9)XY,YXwhere the injection i:XY is continuous and its adjoint, i:YX, is a densely injective. Let {S(t)}t0 be a semigroup on X and Y, respectively, and be a continuous semigroup or a weak continuous semigroup on Y. Then for any bounded subset B of X, {S(t)}t0 is norm-to-weak continuous on S(B).

Theorem 6 (see [<xref ref-type="bibr" rid="B17">16</xref>–<xref ref-type="bibr" rid="B19">18</xref>]).

Assume that {S(t)}t0 is a semigroup on Banach space X and satisfies that

(i) {S(t)}t0 has a bounded absorbing set B0 in X;

(ii) {S(t)}t0 satisfies condition (C) or {S(t)}t0 is ω-limit compact in X.

And assume furthermore that {S(t)}t0 is norm-to-weak continuous on S(B0). Then {S(t)}t0 has a global attractor A in X; i.e., A is nonempty, invariant, compact in X and attracts every bounded subset of X in the norm topology of X.

3. A Priori Estimate

Next we iterate some main results in , which are important for getting a priori estimate.

For any initial data u0v1,u1H, problem (1) possesses a unique weak solution u, which satisfies uC(R+;V1),utC(R+;H), and, for any t0,(10)ξuE12Qξu0E1+Qf.

Furthermore problem (1) generates a dynamical system of solution within the space H02(Ω)×L2(Ω). This system possesses a compact global attractor A, which is bounded in [H4(Ω)H02(Ω)]×H02(Ω).

Choosing 0<ε<2/3, taking the scalar product in H of the first equation of (1) with Av=Aut+εAu, we find(11)ddtξuE22+2εut,Au-2f,Au+2φu,Au+2utV12-2εut,Aut+2εut,Au+2εuV22+2εφu,Au-2εf,Au-2φuut,Au=0.

Owing to  and Ho¨lder inequality, we obtain(12)2φuut,AuφuutAuεuV22+QR,where R=Q(ξu(0)E1)+Q(f). Furthermore we see from (11) that(13)ddtξuE22+2εut,Au-2f,Au+2φu,Au+2-2εutV12+2εut,Au+εuV22+2εφu,Au-2εf,AuQR.We denote the energy functions as follows:(14)E=ξuE22+2εut,Au-2f,Au+2φu,Au.Combining with (13), (12), and 0<ε<2/3, we get(15)dEdt+εEQR.

By the Gronwall inequality, it follows that(16)EEξt0e-εt-t0+FRε1-e-εt-t0,tt0.

Next, we show that(17)12ξuE22-QR,εE32ξuE22+QR,ε.By Ho¨lder inequality, Young’s inequality, Poincaré inequality, and , we conclude that(18)EξuE22+2εutV1uV1+2fuV2+2φuuV2ξuE22+12ξuE22+QR,ε=32ξuE22+QR,ε,and(19)EξuE22-2εcutV1Au-2fAu-2φuAuL2=ξuE22-2εc14Au2+utV12-2εc38Au2-2εc38Au2-QR,ε.=1-2εcξuE22-QR,ε>12ξuE22-QR,ε.Select ε small enough to verify that(20)1-2εc>12.

From (16) and (17), we have(21)ξuE222E+QR,ε2Eξt0e-εt-t0+FRε1-e-εt-t0+QR,ε232ξut0E22+QR,εe-εt-t0+FRε+QR,ε3ξut0E22e-εt-t0+QR,ε.

Now if BBE2(P0,ρ), the ball of E2, center at P0 of radius ρ, then ξu(t0)E2ρ, provided (22)t-t0>1εln3ρ2.So(23)ξuE221+QR=μ.We denote μ=1+Q(R).

4. Existence of the Strong Solution Theorem 7.

Suppose fL2(Ω),φ is a C2(R) function from R into R satisfying (2)-(3) and φ(0)=0. Then given T>0, problem (1) has a unique solution u(x,t) with(24)uL0,T;DA,utL0,T;V1for u0D(A),u1V1. Moreover(u,ut) is a weakly continuous function from [0,T] to E2.

Proof.

We prove the existence of strong solutions by using the Faedo-Galerkin schemes. Assume that there exists an orthonormal basis of D(A) consisting of eigenvectors ωi of A in D(A); simultaneously they are also orthonormal basis of V. The corresponding eigenvalues are λi,i=1,2,, satisfying(25)Aωi=λiωi,iN.

For this purpose, according to the basis theory of ordinary differential equations, we build the sequence of Galerkin approximate solutions. They are smooth functions of the form (26)uN=i=1NμNiωi,and it satisfies(27)ttuN+PNtuN+AuN+PNφuN=PNf,uN0=PNu0,tuN0=PNu1,where Pm:D(A)Vm is the orthogonal projector in Vm, and Vm=span{ω1,ω2ωm}. Of course for every N there is the unique solution uN to (27).

Choosing 0<ε<1, after multiplying (27) by AVm=Aum+ε0Aum, the similar process of (11) leads to(28)ddtξuNE22+2εuNt,AuN-2f,AuN+2φuN,Au+2uNtV12-2εuNt,AuNt+2εuNt,AuN+2εuNV22+2εφuN,AuN-2f,AuN-2φuNuNt,AuN=0.

Like the estimates of (12)-(21), we have(29)ξuNE221+QR.

It means that the sequence {ξuN(t)} is bounded in the space L(0,T;D(A)) for an arbitrary fixed T>0. From (27) we know that(30)d2uNdt=-PNduNdt-AuN-PNφuN+PNf.

So {ttuN} is uniformly bounded in L(0,T;H); then we can extract a subsequences, still denoted by uN, such that(31)uNuinL20,T;DA,uNuinL20,T;H2,uNuinL20,T;H,φuNφuinL20,T;H,PNffinL20,T;H,asN.

It is easy to pass the limit in (27) and we obtain that u is a solution of (1), such that(32)utL0,T;DA,uttL0,T;V1.

Furthermore, from Theorem 4 and , we know that u is a weakly continuous function from [0,T] to E2.

Finally, uniqueness is followed from , since any strong solution would be a weak solution.

Thus, the dynamical system generated by (1) can be defined in the phase space E2, and the corresponding solution semigroup is {s(t)}t>0. By Theorem 7, we have the following results.

Theorem 8.

Suppose the conditions of Theorem 7 hold; then there exists a bounded absorbing set B in E2 for the semigroup {S(t)}t>0.

5. Global Attractors in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M188"><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

We first prove the following compactness results and the norm-to-weak continuity of semigroup.

Lemma 9.

Suppose that (2) and (3) hold; φ(0)=0,φ(u):D(A)V1 is defined by(33)φu,v=Ωφuvdx,uD(A),vH02(Ω). Then φ is continuous compact.

Proof.

Suppose that un is bounded sequences in D(A); without lose of generality, we assume that un weakly converge to u0 in D(A). By the Sobolev embedding theorem, we know that un is bounded and converges to u0 in LP(0,T),W1,P(0,T),W2,P(0,T),P1,T>0. Denote un-u0=ωn; by the results of  and the Sobolev embedding theorem, we show that φ(u),φ(u),φ(u) are uniformly bounded in L; that is, there exists a constant M>0, such that(34)φuLM,φuLM,φuLM,

since there exists constant 0<θ<1, such that(35)0L2φun-φux22dx1/20Lφu0+θωn2u0+θωnx2ωn2dx1/2+0Lφu0+θωnu0+θωnx2ωn2dx1/2+0Lφu0+θωn2ωnx22dx1/2+20Lφu0+θωnu0+θωnxωnx2dx1/2M0L2u0+θωnx2ωn2dx1/2+M0Lu0+θωnx2ωn2dx1/2+M0L2ωnx22dx1/2+2M0Lu0+θωnxωnx2dx1/2

Because un converges to u0 in LP(0,T),W1,P(0,T),W2,P(0,T),P1,T>0, we have(36)limn0L2x2φun-φu02dx1/2=0The proof is completed.

Similarly, we can prove the following Lemma.

Lemma 10.

Let g(u,ut)=φ(u)ut and (2) and (3) hold, φ(0)=0. Then g:D(A)×V1H) and φ(u):D(A)H are continuous compact.

Since D(A)×V1V1×L2, from Theorem 5 and , we can immediately obtain the following result.

Lemma 11.

The semigroup {S(t)}t0 associated with (1) is norm-to weak continuous on S(B), where B is abounded absorbing set of {s(t)}t0 in E2 and S(B) is the stationary set of B defined by(37)SB=xBStxB,t0.

Now we give our main results of the paper.

Theorem 12.

Suppose that fL2(Ω),φC3(R,R),φ(0)=0, and (2) and (3) hold. Then the solution semigroup {s(t)}t0 of problem (1) has global attractor A in E2; it attracts all bounded subset of E2 in the norm of E2.

Proof.

Applying Theorems 7 and 8, we only need to verify that {S(t)}t0 satisfies condition (C) in E2.

Let λ1,λ2, be the eigenvalues of A in D(A) and ω1,ω2, be the corresponding eigenvectors such that (38)0<λ1λ2λn,where λn, as n and {ω1,ω2,} forms an orthogonal basis in D(A) and H02(Ω).

Let Vm=span{ω1,ω2,ωm} and Pm be the canonical projector on Vm and I be the identity. Then, for any (u,ut)E2, it has a unique decomposition  (u,ut)=(u1,u1t)+(u2,u2t), where (39)u,ut=Pmu,Pmut,u2,u2t=I-Pmu,ut.

Since fL2(Ω) and φ:D(A)H(Ω) are compact continuously verified by Lemma 10, then, for any ε>0, there exists N>0, such that, for any m>N, we have(40)I-Pmf<ε8,I-Pmf<ε8,I-Pmφu<ε8,uBV20,μ,where μ is given by (23).

Multiplying (1) by Av=Au2t+αAu2, we can get(41)ddtξu2E22+2αu2t,Au2+2φu,Au2+2u2tV12-2αu2t,Au2t+2αu2t,Au2+2αu2V22+2αφu,Au2=2f,Au2t+2φuu2t,Au2+2αf,Au2.

Applying the Young inequality, Ho¨lder inequality, Sobolev embedding theorem, and (40) and (41), the three terms in the right-hand side of (41) can be estimated as follows: (42)2f,Au2t2f,Au2t2f2V1u2tV1ε4u2tH2u2tH22+C2ε2;2φu,Au22φuut,Au22·ε8u2V2α2u2V22+C1ε2;2αf,Au22αf,Au22f2V1u2V12Cα·ε8u2V2α2u2V22+C3ε2.

Using the above estimates, we transform (41) as follows:(43)ddtξu2E22+2αu2t,Au2+2φu,Au2+2αu2t,Au2+αu2V22+1-2αu2tV12+2αφu2,Au2Cε2,C=C1+C2+C3.

Choose α small enough such that 1-2α>α and 2α>2α2. Hence (43) can be rewritten as (44)ddtξu2E22+2αu2t,Au2+2φu,Au2+α[ξu22+2αu2t,Au2+2φu2,Au2Cε2.

Denote Y(t)=ξu2E22+2α(u2t,Au2)+2(φ(u),Au2) we arrive at (45)dYtdt+αYtCε2.

By Gronwall inequality(46)YtYt1e-αt-t1+Cε2α1-e-αt-t1Yt1e-αt-t1+Cε2α.

Next, we show that(47)12ξu2E22-C4ε2Yt32ξu2E22+C4ε2.

Indeed, the right inequality is obtained using Ho¨lder inequality, Young inequality, and Lemma 9: (48)Ytξu2E22+2αu2tV1u2V1+2φu2Au2ξu2E22+2αεu2tV12+2εAu2ξu2E22+12u2tV12+12Au2+2α2ε2+2ε232ξu2E22+C4ε2.and the left one is the same, where C4=2(1+α2).

Thus, combining (46) and (47) and Theorem 8, we deduce (49)ξu2E222Yt+2C4ε2232R32+C4ε2e-αt-t1+2Cε2α+2C4ε23R32e-αt-t1+C5ε2.

Taking t-t1>t(R3), it follows that (50)ξu2E221+C5ε2,where, by (29), we denote R3=1+Q(R), C5=C/α+C4,t(R3)=t1+1/αln3R32.

Together with Theorems 5, 7, and 8, the proof is finished.

Remark 13.

If we transform the first equation of (1) to the following form, (51)utt+σuut+2u+φu=f,then the term σ(u)ut accounts for dynamical friction, and we only need to assume the damping coefficient σ(u) is a positive function and by adding some appropriate conditions of continuity, conclusions of Theorem 12 remain valid and the proof’s process has no essential difference.

Data Availability

All data included in this study are available upon request by contact with the corresponding author.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by the National Science Foundation of China (61703181).

Mao A. Chang H. Kirchhoff type problems in RN with radial potentials and locally Lipschitz functional Applied Mathematics Letters 2016 62 49 54 10.1016/j.aml.2016.06.014 MR3545337 Mao A. Zhu X. Existence and multiplicity results for Kirchhoff problems Mediterranean Journal of Mathematics 2017 14 2 Art. 58, 14 10.1007/s00009-017-0875-0 MR3619420 Zbl1371.35059 Gao Q. Li F. Wang Y. Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation Central European Journal of Mathematics 2011 9 3 686 698 10.2478/s11533-010-0096-2 MR2784038 Zbl1233.35145 2-s2.0-79953292078 Kolbasin S. Attractors for Kirchhoff's equation with a nonlinear damping coefficient Nonlinear Analysis 2009 71 5-6 2361 2371 10.1016/j.na.2009.01.187 MR2524443 Khanmamedov A. K. Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain Journal of Differential Equations 2006 225 2 528 548 10.1016/j.jde.2005.12.001 MR2225799 Zbl1101.35019 2-s2.0-33646364027 Khanmamedov A. K. A global attractor for the plate equation with displacement-dependent damping Nonlinear Analysis 2011 74 5 1607 1615 10.1016/j.na.2010.10.031 MR2764362 Zhijian Y. Yunqing W. Global attractor for the Kirchhoff type equation with a strong dissipation Journal of Differential Equations 2010 249 12 3258 3278 10.1016/j.jde.2010.09.024 MR2737429 Zbl1213.35126 2-s2.0-78049279302 Yang Z. Li X. Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation Journal of Mathematical Analysis and Applications 2011 375 2 579 593 10.1016/j.jmaa.2010.09.051 MR2735547 Zbl1213.35125 2-s2.0-78149414068 Chueshov I. Long-time dynamics of Kirchhoff wave models with strong nonlinear damping Journal of Differential Equations 2012 252 2 1229 1262 10.1016/j.jde.2011.08.022 MR2853537 Zbl1237.37053 2-s2.0-80655149011 Li F. Zhao C. Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping Nonlinear Analysis 2011 74 11 3468 3477 10.1016/j.na.2011.02.033 MR2803074 Li F. Zhao Z. Chen Y. Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation Nonlinear Analysis: Real World Applications 2011 12 3 1759 1773 10.1016/j.nonrwa.2010.11.009 MR2781894 Zbl1218.35040 Li F. Limit behavior of the solution to nonlinear viscoelastic Marguerre -von Karman shallow shells system Journal of Differential Equations 2010 249 6 1241 1257 10.1016/j.jde.2010.05.005 MR2677793 Sun M. Bai Q. A new descent memory gradient method and its global convergence Journal of Systems Science & Complexity 2011 24 4 784 794 10.1007/s11424-011-8150-0 MR2835356 Zhong C.-K. Yang M.-H. Sun C.-Y. The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations Journal of Differential Equations 2006 223 2 367 399 10.1016/j.jde.2005.06.008 MR2214940 Zbl1101.35022 2-s2.0-33645129548 Ma Q. Wang S. Zhong C. Necessary and sufficient conditions for the existence of global attractors for semigroups and applications Indiana University Mathematics Journal 2002 51 6 1541 1559 10.1512/iumj.2002.51.2255 MR1948459 Zbl1028.37047 Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics 1997 68 2nd New York, NY, USA Springer Applied Mathematical Sciences 10.1007/978-1-4612-0645-3 MR1441312 Zbl0871.35001 Ma Q. Zhong C. Existence of strong solutions and global attractors for the coupled suspension bridge equations Journal of Differential Equations 2009 246 10 3755 3775 10.1016/j.jde.2009.02.022 MR2514725 Zbl1176.35036 Robinson J. C. Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative parabolic PDEs and the Theory Of Global Attractors 2001 Cambridge, UK Cambridge University Press Cambridge Texts in Applied Mathematics MR1881888 Zbl0980.35001