# Strong Solutions and Global Attractors for Kirchhoff Type Equation

Authors: Xiangping Chen
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 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,u∂Ω=△u∂Ω=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 φ∈C1(R) with φ(0)=0 fulfills the dissipation inequality(2)liminfu→∞⁡φuu>-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,w∂Ω=0.Whenn=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 [1–3] and the reference therein.When the coefficient ofut 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 [4]. 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 [5–9], 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 [5]. For Ω a three-dimensional unbounded domain and under suitable conditions on σ(u) and f(u), A.Kh. Khanmamedov in [6] 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 [7–9] 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<δ≤1.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 [9–14]. Furthermore, the global attractor is established in the higher energy space in H4(Ω)∩H02(Ω).The paper is organized as follows. In Section2, 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=H2Ω∩H01Ω,V2=DA=u∈H2Ω∣Au∈L2Ω,whereA=△2.The norm and the scalar product inL2(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 functionu(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 lettersC 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 [14–18] and the references therein for more details.Definition 1 (see [16, 18]).
LetX be a Banach space and {S(t)}t≥0 be a family operator on X. We say that {S(t)}t≥0 is a norm-to-weak continuous semigroup on X, if {S(t)}t≥0 satisfies(i)
S(0)=Id (the identity);(ii)
S(t)S(s)=S(t+s), ∀t,s≥0;(iii)
S(tn)xn⇀S(t)x, if tn→t,xn→x in X.Definition 2 (see [15]).
AC0 semigroup {S(t)}t≥0 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,x∈B,t≥t(B)} is bounded and(8)I-PStxX<ε,t≥tB,x∈B,where P:X→X1 is a bounded projector.Theorem 3 (see [14]).
Let X be a Banach space and{S(t)}t≥0 be a norm-to-weak continuous semigroup on X. Then {S(t)}t≥0 has a global attractor in X provided that the following conditions hold true:
(i){S(t)}t≥0 has a bounded absorbing set B0 in X.
(ii){S(t)}t≥0 satisfies the condition (C).Theorem 4 (see [16, 18]).
Let X and Y be two Banach spaces such thatX⊂Y 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 [16, 18]).
Let X, Y be two Banach spaces andX∗,Y∗ be their dual spaces, respectively, such that(9)X↪Y,Y∗↪X∗where the injection i:X→Y is continuous and its adjoint, i∗:Y∗↪X∗, is a densely injective. Let {S(t)}t≥0 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)}t≥0 is norm-to-weak continuous on S(B).Theorem 6 (see [16–18]).
Assume that{S(t)}t≥0 is a semigroup on Banach space X and satisfies that
(i){S(t)}t≥0 has a bounded absorbing set B0 in X;
(ii){S(t)}t≥0 satisfies condition (C) or {S(t)}t≥0 is ω-limit compact in X.
And assume furthermore that{S(t)}t≥0 is norm-to-weak continuous on S(B0). Then {S(t)}t≥0 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 [4], which are important for getting a priori estimate.For any initial datau0∈v1,u1∈H, problem (1) possesses a unique weak solution u, which satisfies u∈C(R+;V1),ut∈C(R+;H), and, for any t≥0,(10)ξuE12≤Qξu0E1+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).Choosing0<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-2φ′uut,Au=0.Owing to [4] and Ho¨lder inequality, we obtain(12)2φ′uut,Au≤φ′uutAu≤εuV22+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,Au≤QR.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+εE≤QR.By the Gronwall inequality, it follows that(16)E≤Eξt0e-εt-t0+FRε1-e-εt-t0,∀t≥t0.Next, we show that(17)12ξuE22-QR,ε≤E≤32ξuE22+QR,ε.By Ho¨lder inequality, Young’s inequality, Poincaré inequality, and [4], we conclude that(18)E≤ξuE22+2εutV1uV1+2fuV2+2φuuV2≤ξuE22+12ξuE22+QR,ε=32ξuE22+QR,ε,and(19)E≥ξuE22-2εcutV1Au-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)ξuE22≤2E+QR,ε≤2Eξt0e-εt-t0+FRε1-e-εt-t0+QR,ε≤232ξut0E22+QR,εe-εt-t0+FRε+QR,ε≤3ξut0E22e-εt-t0+QR,ε.Now ifB⊂BE2(P0,ρ), the ball of E2, center at P0 of radius ρ, then ξu(t0)E2≤ρ, provided (22)t-t0>1εln⁡3ρ2.So(23)ξuE22≤1+QR=μ.We denote μ=1+Q(R).


## 4. Existence of the Strong Solution

Theorem 7.
Supposef∈L2(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)u∈L∞0,T;DA,ut∈L∞0,T;V1for u0∈D(A),u1∈V1. 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 ofD(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,∀i∈N.
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+PN∂tuN+AuN+PNφuN=PNf,uN0=PNu0,∂tuN0=PNu1,where Pm:D(A)→Vm is the orthogonal projector in Vm, and Vm=span{ω1,ω2⋯ωm}. Of course for every N there is the unique solution uN to (27).
Choosing0<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-2φ′uNuNt,AuN=0.
Like the estimates of (12)-(21), we have(29)ξuNE22≤1+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)uN⇀uinL20,T;DA,uN′⇀u′inL20,T;H2,uN′′⇀u′′inL20,T;H,φuN⇀φuinL20,T;H,PNf⇀finL20,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)ut∈L∞0,T;DA,utt∈L∞0,T;V1.
Furthermore, from Theorem4 and [4], we know that u is a weakly continuous function from [0,T] to E2.
Finally, uniqueness is followed from [4], 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 Theorem7 hold; then there exists a bounded absorbing set B in E2 for the semigroup {S(t)}t>0.


## 5. Global Attractors inE2

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=∫Ω△φu△vdx,∀u∈D(A),v∈H02(Ω). Then φ is continuous compact.Proof.
Suppose thatun 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),∀P≥1,∀T>0. Denote un-u0=ωn; by the results of [4] 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)φ′uL∞≤M,φ′′uL∞≤M,φ′′′uL∞≤M,
since there exists constant0<θ<1, such that(35)∫0L∂2φun-φu∂x22dx1/2≤∫0Lφ′′u0+θωn∂2u0+θωn∂x2ωn2dx1/2+∫0Lφ′′′u0+θωn∂u0+θωn∂x2ωn2dx1/2+∫0Lφ′u0+θωn∂2ωn∂x22dx1/2+2∫0Lφ′′u0+θωn∂u0+θωn∂x∂ωn∂x2dx1/2≤M∫0L∂2u0+θωn∂x2ωn2dx1/2+M∫0L∂u0+θωn∂x2ωn2dx1/2+M∫0L∂2ωn∂x22dx1/2+2M∫0L∂u0+θωn∂x∂ωn∂x2dx1/2
Becauseun converges to u0 in LP(0,T),W1,P(0,T),W2,P(0,T),∀P≥1,∀T>0, we have(36)limn→∞⁡∫0L∂2∂x2φun-φu02dx1/2=0The proof is completed.Similarly, we can prove the following Lemma.Lemma 10.
Letg(u,ut)=φ′(u)ut and (2) and (3) hold, φ(0)=0. Then g:D(A)×V1→H) and φ′(u):D(A)→H are continuous compact.SinceD(A)×V1↪V1×L2, from Theorem 5 and [4], we can immediately obtain the following result.Lemma 11.
The semigroup{S(t)}t≥0 associated with (1) is norm-to weak continuous on S(B), where B is abounded absorbing set of {s(t)}t≥0 in E2 and S(B) is the stationary set of B defined by(37)SB=x∈B∣Stx∈B,∀t≥0.Now we give our main results of the paper.Theorem 12.
Suppose thatf∈L2(Ω),φ∈C3(R,R),φ(0)=0, and (2) and (3) hold. Then the solution semigroup {s(t)}t≥0 of problem (1) has global attractor A in E2; it attracts all bounded subset of E2 in the norm of E2.Proof.
Applying Theorems7 and 8, we only need to verify that {S(t)}t≥0 satisfies condition (C) in E2.
Letλ1,λ2,⋯ be the eigenvalues of A in D(A) and ω1,ω2,⋯ be the corresponding eigenvectors such that (38)0<λ1≤λ2≤⋯≤λn≤⋯,where λn→∞, as n→∞ and {ω1,ω2,⋯} forms an orthogonal basis in D(A) and H02(Ω).
LetVm=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.
Sincef∈L2(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-Pmφ′u<ε8,∀u∈BV20,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+2φ′uu2t,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,Au2t≤2f,Au2t≤2f2V1u2tV1≤ε4u2tH2≤u2tH22+C2ε2;2φ′u,Au2≤2φ′uut,Au2≤2·ε8u2V2≤α2u2V22+C1ε2;2αf,Au2≤2αf,Au2≤2f2V1u2V1≤2Cα·ε8u2V2≤α2u2V22+C3ε2.
Using the above estimates, we transform (41) as follows:(43)ddtξu2E22+2αu2t,Au2+2φu,Au2+2αu2t,Au2+αu2V22+1-2αu2tV12+2αφu2,Au2≤Cε2,C=C1+C2+C3.
Chooseα 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,Au2≤Cε2.
DenoteY(t)=ξu2E22+2α(u2t,Au2)+2(φ(u),Au2) we arrive at (45)dYtdt+αYt≤Cε2.
By Gronwall inequality(46)Yt≤Yt1e-αt-t1+Cε2α1-e-αt-t1≤Yt1e-αt-t1+Cε2α.
Next, we show that(47)12ξu2E22-C4ε2≤Yt≤32ξu2E22+C4ε2.
Indeed, the right inequality is obtained usingHo¨lder inequality, Young inequality, and Lemma 9: (48)Yt≤ξu2E22+2αu2tV1u2V1+2φu2Au2≤ξu2E22+2αεu2tV12+2εAu2≤ξu2E22+12u2tV12+12Au2+2α2ε2+2ε2≤32ξu2E22+C4ε2.and the left one is the same, where C4=2(1+α2).
Thus, combining (46) and (47) and Theorem 8, we deduce (49)ξu2E22≤2Yt+2C4ε2≤232R32+C4ε2e-αt-t1+2Cε2α+2C4ε2≤3R32e-αt-t1+C5ε2.
Takingt-t1>t(R3), it follows that (50)ξu2E22≤1+C5ε2,where, by (29), we denote R3=1+Q(R), C5=C/α+C4,t(R3)=t1+1/αln⁡3R32.
Together with Theorems5, 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.



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