ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR A SEMIBOUNDED NONMONOTONE EVOLUTION EQUATION

FOR A SEMIBOUNDED NONMONOTONE EVOLUTION EQUATION

NIKOS KARACHALIOS, NIKOS STAVRAKAKIS, AND PAVLOS XANTHOPOULOS

Received 3 September 2002

We consider a nonlinear parabolic equation involving nonmonotone diffusion.

Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global at- tractor for the corresponding dynamical system.

1. Introduction

We consider the following nonlinear parabolic initial boundary value problem in the open bounded intervalΩ⊂R

u_t−a(u)u_xx−b(u)u²_x−λσ(u)= f(x), x∈Ω, t >0, (1.1a)

u(x,0)=u0(x), (1.1b)

u|∂Ω=0, t >0. (1.1c) This problem extends the well studied porous medium diffusion, since no cer- tain relationship between the coefficientsa(u) andb(u) is assumed. Let us men- tion that special cases of this system may typically arise in plasma physics within the context of the fluid treatment of charged particles, and in density-dependent reaction diffusion processes in mathematical biology. Naturally enough, these systems imply only positive values foru(x, t); however, in the following treat- ment, wedo notimpose such a restriction.

In order to demonstrate a specific case-modelled system, we consider the col- lisionless evolution equation for the electron pressureP=nT, which, if we ig- nore viscosity, gets the following form in thex-direction (see, Balescu [5])

3

2Pt= −qx−3 2uPx−5

2Pux, (1.2)

Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:9 (2003) 521–538 2000 Mathematics Subject Classification: 35K55, 35B40, 35B41 URL:http://dx.doi.org/10.1155/S1085337503210022

whereurepresents the electron velocity andq is the heat flux. Now, applying Darcy’s law (see, Aronson [3])

u= −cPx, c >0, (1.3) to the above equation, we get

3

2P_t= −q_x+3 2cP_x²+5

2cPP_xx. (1.4)

We see that the first term on the right-hand side corresponds to porous me- dium diffusion (not considered here), whereas the other two terms constitute a specific case of (1.1a), witha(P)=(5/3)cPandb(P)=c.

Concerning the applications in the dynamics of cell populations, with a spa- tial distribution of cells, we may associate an energy densitye(u), that is an inter- nal energy per unit volume of an evolving spatial pattern, whereu(x, t) denotes the cell density (see [6,14]). In this case, the total energyE(u) in a volumeVis given by

E(u)=

Ve(u)dx. (1.5)

The change in energyδE, that is the work done in changing states by an amount δu, is given by the variational derivativeδE/δuwhich defines a potential

µ(u)=δE

δu⁼e(u). (1.6)

The gradient of the potentialµproduces a fluxJ, which is classically proportional to this gradient, that is

J= −kµ(u). (1.7)

By using (1.6) and (1.7), the continuity equation for the densityuis

∂u

∂t ⁼

a(u)u_xx, a(u)=ke(u). (1.8) Writing out the diffusion term in full, we end up with the nonlinear operator that appears in (1.1a), in the special case where it holdsa(u)=b(u), that is the porous medium case. Also, the nonlinearityσ(u) may stand for the possible growth dynamics.

For completeness, let us mention some of the results, concerning the large time behavior of bounded solutions of nonlinear diffusion equations. Most of them are related to porous medium type equations (degenerate, monotone dif- fusion). In [4], the existence of a global attractor for the one-dimensional porous medium equation, attracting all orbits starting fromL^∞-initial data, is demon- strated. Extensive studies in [1,13,15] show that theω-limit set is contained

in the set of stationary solutions. Extensions for the unbounded domain case can be found in [10,11]. We also mention [2,7,8] on the existence of global attractors for degenerate or nondegenerate quasilinear parabolic equations.

The principal assumption that will be used throughout this paper in the study of problem (1.1) is the following assumption.

Hypothesis 1.1. a, b, σ∈C²(R),λ∈R, and there existsc_∗>0 such thata(s)≥c_∗ (i.e., we consider nondegenerate butnonmonotonediffusion).

Due to the nonmonotonicity, the standard compactness methods on exis- tence of solutions are not sufficient. To this end,the diffusion operator is treated as a semibounded operator within the functional setting of an admissible triple. This procedure allows for the construction of unique solutions in Cw([0, T], H²∩ H₀¹(Ω)), the space of weakly continuous functionsu: [0, T]→H²∩H₀¹(Ω).

The existence of a global attractor in the phase space H=H²∩H₀¹(Ω) is proved inSection 3. The result is shown assuming monotonicity for the non- linearityb(·), considered to be nonincreasing.Nevertheless, this assumption does not imply monotonicity for the diffusion operator itself. An important feature is that this assumption is sufficient to prove further regularity with respect to time for the solutions of (1.1) constructed inSection 2. Further, using this result, we may define the semigroupS(t) :u0∈H →u(t)∈H, corresponding to our prob- lem.

We conclude by recalling some well-known results, which will be frequently used (see, [16,17,18,19]).

Lemma1.2 (Gagliardo-Nirenberg inequality). Let 1≤p, q, r≤ ∞, j an integer, 0≤j≤m, and j/m≤θ≤1. Then

D^ju_p≤constu¹_q^−θD^mu^θ_r, u∈L^q∩W^m,r(Ω),Ω⊆Rⁿ, (1.9)

where

1 p ⁼

j n+θ

1 r⁻

m n

+1−θ

q . (1.10)

Ifm−j−n/ris not a nonnegative integer, then the inequality holds forj/m≤θ <1.

Lemma 1.3 (uniform Gronwall). Let g, h, y be three positive locally integrable functions fort0≤t <∞which satisfy

d y

dt ^≤g y+h, ∀t≥t0, _t+r

t g(s)ds≤α1,

_t+r

t h(s)ds≤α2,

_t+r

t y(s)ds≤α3,

(1.11)

for allt≥t0, whereα1,α2, andα3are positive constants. Then y(t+r)≤

α3

r +α2

expα1

, ∀t≥t0. (1.12)

We also use the short (equivalent) normsu_x2,u_xx2, andu_xxx2inH₀¹(Ω), H²∩H0¹(Ω), andH³∩H0¹(Ω), respectively (seeSection 3). From the embedding H^k∩H0¹(Ω)C_b^k−1(Ω),k=1,2, . . . ,and the Poincar´e inequality (see [9, page 242]), we have

u^(k−1)_∞≤constu_H^k_∩H0¹≤constu^(k)₂. (1.13)

2. Local existence

To obtain results on local existence of solutions, we intend to write problem (1.1) as a nonlinear evolution equation in an appropriate functional setting. More precisely, we will consider anadmissible tripleof Banach spaces, which is defined as follows (see [17,18] and [19, page 784]).

Definition 2.1. Anadmissible tripleVHW has the following properties:

(i)His a real separable Hilbert space with scalar product (·|·)H, (ii){V, W}is a dual pairof real separable Banach spaces with the corresponding bilinear form

·,·(i.e.,·,·is continuous,w, v =0, for everyw∈W, impliesv=0, and w, v =0, for everyv∈V, impliesw=0), (iii) the embeddingsVHW are continuous and dense, (iv) it holdsh, v =(h|v)_H, for allh∈H,v∈V.

Clearly, an admissible triple generalizes the notion of the evolution triple, in the sense that for an admissible triple it may holdW=V^∗. This generalization is necessary in order to tackle the extended version of diffusion in hand. For problem (1.1), we select the spaces

V=H⁴∩H₀¹(Ω), H=H²∩H₀¹(Ω), W=L²(Ω). (2.1) Lemma2.2. The embeddingVHWfor the spaces (2.1) defines an admissible triple.

Sketch of the proof. Consider the bilinear form ·,·:W×V →R, defined by the integral

w, v =

Ωvw+wv_xxxxdx, ∀v∈V, w∈W. (2.2) Now, it is easy to check that the inner product stemming from the bilinear form·,·

(w|v)H=

Ωvw+wxxvxxdx, for everyw∈H, v∈H (2.3)

induces an equivalent norm inH. We also have that w, v =

Ωvw+wv_xxxxdx

≤ w2v2+w2vxxxx

2

≤cwWvV,

(2.4)

hence the bilinear form ·,·is continuous. Now assume that, for some w∈ W, it holdsw, v =0, for everyv∈V. Classical arguments on existence and regularity of solutions for linear elliptic equations (see [12, Chapter II]) imply the existence of solutions for the problem

v−vxxxx=w, v∈V. (2.5)

For this solutionv, we have that

0= w, v =

Ωw²dx, (2.6)

which implies thatw=0 and the proof is complete.

We introduce the nonlinear operatorsA,B:V →Wdefined by

Au= −a(u)u_xx, Bu= −b(u)u²_x. (2.7) The following results outline the basic properties of the operatorsAandB.

Proposition2.3. The operatorA+B:H →Wis bounded on bounded sets ofH. Proof. LetB=BH(R) be a closed ball inH. We will show that there exist con- stantsK1(R) andK2(R) such that

Au2≤K1(R)uH, Bu2≤K2(R)uH, ∀u∈B. (2.8) Sincea,b,σ∈C²(R) and the embeddingHC¹_b(Ω) is continuous, it follows that there exist constantsC_1,m(R) andC_2,m(R),m=0,1,2, such that

sup

x∈Ω

a^(m)u(x)≤C_1,m(R), m=0,1,2, (2.9) sup

x∈Ω

b^(m)u(x)≤C_2,m(R), m=0,1,2. (2.10)

Using (2.9), (2.10), and the fact thatH0¹(Ω) is a generalized Banach algebra, we may obtain the inequalities

Au2≤sup

x∈Ω

au(x)uxx

2≤K1(R)uH, Bu2≤sup

x∈Ω

bu(x)u²_x2≤const sup

x∈Ω

bu(x)u²_H

≤K2(R)uH.

(2.11)

Finally, we conclude that

(A+B)u₂≤K(R)uH, (2.12)

whereK(R)=max{K1(R), K2(R)}.

Proposition2.4. The operatorA+B:H →Wis locally Lipschitz continuous.

Proof. Letu, v∈B=B_H(R) be a closed ball inH. We have that Au−Av2≤a(u)−a(v)vxx

2+a(u)uxx−vxx

2. (2.13)

From the mean value theorem and (2.9), we get

au(x)−av(x)≤C1,1(R)u(x)−v(x), (2.14) au(x)−av(x)≤C1,2(R)u(x)−v(x). (2.15)

Therefore,

a(u)−a(v)vxx²

2≤C1,1(R)²u−v²_∞vxx²

2≤C(R)u−v²_H, a(u)uxx−vxx²

2≤C²_1,0(R)uxx−vxx²

2≤C(R)u−v²_H, (2.16) whereC(R) is a common symbol for the constants. Similar inequalities hold for the operatorB. So finally it holds that

(A+B)u−(A+B)v₂≤C(R)u−vH. (2.17) Proposition2.5. The operatorA+B:H →Wis semibounded.

Proof. By definition, it must be proved that there exists a monotone increasing functiond1∈C¹(R) such that

(A+B)u, u ≥ −d1

u²_H

, for everyu∈V. (2.18) Letu∈C^∞0(Ω)∩C(Ω). For the operatorA, it holds

Au, u =

ΩAuu dx+

ΩAuuxxxxdx. (2.19)

Integration by parts in the second integral on the right-hand side of (2.19) gives

−

Ωa(u)uxxuxxxxdx= −1 2

Ωa(u)u²_xu²_xxdx−1 2

Ωa(u)u³_xxdx +

Ωa(u)u²_xxxdx.

(2.20)

UsingLemma 1.2, we obtain the inequality

u_xx4≤constu^1/42 u_xxx^3/4₂ , (2.21)

which, with the aid of (2.9) and Young’s inequality, gives the following estimate:

−1 2

Ωa(u)u²_xu²_xxdx−1 2

Ωa(u)u³_xxdx

≥ −C1,2ux²

∞uxx²

2−C1,1uxx

2uxx²

4

≥ −Cˆ1u⁴_H−Cˆ2uHu^1/22 uxxx^3/2

2

≥ −Cˆ1u⁴_H−Cˆ3u^3/2_H uxxx^3/2

2

≥ −Cˆ1u⁴_H−Cˆ4u⁶_H−c_∗ 2 uxxx²

2.

(2.22)

For the first integral of the right-hand side of (2.19), we have

−

Ωa(u)uxxu dx≥ −C1,0u∞uxx1≥ −Cˆ0u²_H. (2.23) UsingHypothesis 1.1, (2.19), (2.20), (2.22), (2.23), and density arguments, we obtain that

Au, u ≥ −Cˆ0u²_H−Cˆ1u⁴_H−Cˆ4u⁶_H:= −d_1,1u²_H

. (2.24)

A similar procedure may be followed for the operatorB, to derive the relation Bu, u ≥ −d1,2

u²_H

. (2.25)

Finally, from estimates (2.24) and (2.25) we get that there exists a monotone increasingC¹- functiond1:R→Rsatisfying (2.18).

The previous propositions enable us to show local existence of solutions. The result is stated as follows.

Theorem2.6. Letu0, f ∈H. Assume thatHypothesis 1.1is satisfied. Then there existsT >0such that problem (1.1) has a unique solution

u∈C_w[0, T], H, u_t∈C_w[0, T], W. (2.26) Moreover, the solutionu: [0, T]→Wis Lipschitz continuous.

Proof. (A)Existence: the first step is to show existence of at least one solution in a finite dimensional subspaceVn=span{e1, . . . , en}ofV, where{ei}i≥1is an orthonormal basis ofVnwith respect to (·|·)H. It holds that_nVn=VH.

We define the linear and continuous operator ˜P_n:W →V as P˜_nw=

n i

w, e_i e_i, w∈W. (2.27)

Now, the Galerkin equation for problem (1.1) onVnVHreads

u_n(t) + ˜P_n(A+B)u_n(t)=P˜_nCu_n(t), t∈[0, T], u_n(0)=P˜_nu0, (2.28) where

Cun(t)=λσun(t)+f . (2.29) Using Propositions2.3and2.4, Peano’s theorem justifies the existence of aC¹ solution for (2.28),un: [0, T0]→Vn, for someT0>0 which depends onn.

The next step is to obtain an a priori estimate for un inH. Note that ˜Pn: H →V_n is an orthogonal projection onto the spaceV_n, since it holds ˜P_nu= _n

i(u|e_i)_He_i,u∈H. Sinceu_nis continuous on [0, T0], (2.28) implies that u_n|un

H= −P˜n(A+B)un|un

H+P˜nCun|un

H

= −

(A+B)un, un +Cun, un . (2.30) Now, it is not hard to verify that there exists a monotone increasing function d2∈C¹(R) such that

Cu, u ≤d2

u²H

, ∀u∈V. (2.31)

Hence, from (2.18), (2.30), and (2.31) we obtain the differential inequality d

dtun(t)²_H≤2dun(t)²_H, t∈ 0, T0

, (2.32)

whereun(0)H= Pnu0H≤ u0H. Since the functiond(·) is Lipschitz con- tinuous as aC¹function, we may apply the theorem of Picard-Lindel¨of to con- clude that there exists aT >0, this time independent ofn, such that

un(t)²_H≤ max

t∈[0,T]g(t)≤R, t∈[0, T]. (2.33) Finally, using standard continuation arguments, we can extend the solutionu_n to the interval [0, T].

Now, from (2.33) we have that there exists a subsequence, denoted again by {u_n},such that

u_n(t) u(t), inH,asn−→ ∞, (2.34) at least in a dense countable subset of [0, T]. Letv∈VkH,k≤n. Since ˜Pnv= v, for everyk≤n, it follows that

u_n(t)|v_H= −P˜n(A+B−C)un(t)|v_H= −

(A+B−C)un(t), v . (2.35) Using Proposition 2.3 and estimate (2.33), we conclude that (un(t)|v)H is equicontinuous on [0, T], which implies that (2.34) holds in the whole inter- val [0, T]. Finally, passing to the limit to (2.35) and using density of_kVkinH, we obtain thatu∈Cw([0, T], H),ut∈Cw([0, T], W) is a solution for problem (1.1) and as a consequence,u: [0, T]→Wis Lipschitz continuous.

(B)Uniqueness: the difference of solutionsw=u−vof problem (1.1) satisfies the following initial value problem:

wt−a(u)wxx−A(u, v)vxx−B(u, v)−λΣ(u, v)=0, w(0)=0, (2.36) where A(u, v)=a(u)−a(v),B(u, v)=(b(u)−b(v))v_x²+b(u)(u²_x−v²_x), andΣ(u, v)=σ(u)−σ(v). Multiplying (2.36) byuand integrating overΩ, we obtain the equation

1 2

d dtw²2+

Ωa(u)v_xww_xdx+

Ω

a(u)−a(v)w_xv_xdx +

Ωa(u)u_xww_xdx+

Ω

a(u)−a(v)v_x²w dx

−

Ω

b(u)−b(v)v_x²w dx−

Ωb(u)u²_x−v²_xw dx +

Ωa(u)w²_xdx−λ

Ω

σ(u)−σ(v)w dx=0.

(2.37)

Using estimate (2.33) and relations (2.9), (2.10), and (2.15) the following es- timates are derived:

Ω

a(u)−a(v)v_x²w dx≤C_1,2v_x²_∞w²2≤C(R)w²2,

Ωa(u)vxwwxdx≤C1,1vx

∞w2wx

2

≤0w_x²₂+C(R)w²2.

(2.38)

The rest of the integrals in (2.37) can be estimated in a similar way. Hence, for sufficiently small0, we get the inequality

1 2

d

dtw(t)²_W+c_∗

2w_x²₂≤Cw(t)²_W. (2.39) Application of the standard Gronwall’s lemma implies uniqueness.

3. Existence of a global attractor inH

In this section, we discuss the asymptotic behavior of solutions of the nonlinear parabolic problem (1.1). To this end, in addition to the principal hypothesis, Hypothesis 1.1, we assume that the nonlinear functionsb,σsatisfy the following hypothesis.

Hypothesis 3.1. b(s)≤0 and there existcm>0, such that|σ^(m)(s)| ≤cm|s|, for allm=0,1,2.

First, we prove that under the extra hypothesis,Hypothesis 3.1, the unique local solutionu(x, t) of problem (1.1), obtained inTheorem 2.6, exists globally in time. We denote byλ_∗the positive constant induced by Poincar´e’s inequality.

Lemma3.2. Let Hypotheses1.1and3.1be fulfilled andu0, f ∈H. Assume also that

λ <c_∗λ_∗ 2c0

. (3.1)

Then there exists a constantρ2independent oft, such that,

lim sup

t→∞

ux(t)₂≤ρ2. (3.2)

Proof. We multiply (1.1a) by−uxxand integrate overΩto get 1

2 d dtux²

2+

Ωa(u)u²_xxdx+

Ωb(u)u²_xuxxdx +λ

Ωσ(u)uxxdx=

Ωf uxxdx.

(3.3)

UsingHypothesis 1.1, we observe that

Ωa(u)u²_xxdx≥c_∗uxx²

2, (3.4)

whereas fromHypothesis 3.1we have

Ωb(u)u²_xuxxdx= −1 3

Ωb(u)u⁴_xdx≥0. (3.5) Furthermore,Hypothesis 3.1, together with Poincar´e’s inequality

u2≤λ⁻_∗^1/2u_x2, (3.6) implies that

λ

Ωσ(u)uxxdx≤λc0u2uxx

2≤λλ⁻_∗¹c0uxx²

2. (3.7)

Relations (3.3), (3.4), and (3.7) imply that d

dtu_x(t)²₂+αu_xx(t)²₂≤ 1

c_∗f²2, (3.8) whereα=c_∗−2c0λλ⁻_∗¹. Applying again Poincar´e’s inequality (3.6) to the above estimate (3.8), we get

d

dtux(t)²₂+αλ_∗ux(t)²₂≤ 1

c_∗f²2. (3.9) If assumption (3.1) is satisfied, that is,α >0 Gronwall’s lemma leads to the following estimate:

ux(t)²₂≤ux(0)²₂exp−αλ_∗t+ 1 αc_∗λ_∗f²2

1−exp−αλ_∗t. (3.10)

Lettingt→ ∞, from estimate (3.10) we obtain that lim sup

t→∞

ux(t)²₂≤ρ²₂, (3.11)

whereρ²₂=(1/αc_∗λ_∗)f²2and the proof is completed.

LetᏮbe a bounded set ofH, included in a ballBH(0, M) ofH, centered at 0 of radiusM. Assuming thatu0∈Ꮾ, we infer fromLemma 3.2that forρ2> ρ2, there existst0(Ꮾ, ρ₂)>0 such that fort≥t0(Ꮾ, ρ₂)

ux(t)₂≤ρ2, u(t)₂≤ρ1=λ⁻_∗^1/2ρ2. (3.12) Integrating (3.8) with respect tot, it follows that for everyr >0

α _t+r

t

u_xx(s)²₂ds≤ r

c_∗f²2+u_x(t)²₂. (3.13) Once again, lettingt→ ∞, we obtain from inequality (3.12) that

lim sup

t→∞

_t+r

t

u_xx(s)²₂ds≤ r

αc_∗f²2+ρ²₂

α, for everyr >0. (3.14) and fort≥t0(Ꮾ, ρ2)

_t+r

t

u_xx(s)²₂ds≤ r

αc_∗f²2+ρ₂²

α , for everyr >0. (3.15) Lemma3.3. Let Hypotheses1.1and3.1be fulfilled,u0∈Ꮾ, and f ∈H. Assume also that (3.1) is satisfied. Then there exists a constantρ3 independent oft, and t1>0such that

uxx(t)₂≤ρ3, fort≥t1. (3.16) Proof. Multiply (1.1a) byuxxxxand integrate overΩto get

1 2

d

dtu_xx²₂+

Ωa(u)u_xu_xxu_xxxdx+

Ωa(u)u²_xxxdx + 2

Ωb(u)uxuxxuxxxdx+λ

Ωσ(u)uxuxxxdx +

Ωb(u)u³_xuxxxdx

= −

Ωfxuxxxdx.

(3.17)

Using inequalities (1.13), (3.12), andHypothesis 1.1, we obtain that inequal- ities (2.9) and (2.10) hold, for allt≥t0(Ꮾ, ρ₂), withRreplaced byρ₂. It follows that

Ωa(u)uxuxxuxxxdx≤C1,1ux

∞uxx

2uxxx

2

≤C1,1constuxx²

2uxxx

2

≤C1uxx⁴

2+1uxxx²

2.

(3.18)

ApplyingLemma 1.2, we obtain the inequality

u_x6≤constu^1/32 u_xx^2/3₂ , (3.19) which can be used to get the estimate

Ωb(u)u³_xuxxxdx≤C2,1ux³

6uxxx

2

≤C2,1constu2uxx²

2uxxx

2

≤C2uxx⁴+1uxxx²

2.

(3.20)

We also have that the estimate λ

Ωσ(u)uxuxxxdx≤λc1u∞ux

2uxxx

2

≤λc1constuxx²

2uxxx

2

≤C3uxx⁴

2+1uxxx²

2.

(3.21)

The rest of the integral terms in (3.17) can be bounded similarly. Thus, for sufficiently small1, we get the inequalities

d

dtuxx(t)²₂+c_∗uxxx(t)²₂≤M1+M2uxx(t)⁴₂, (3.22) d

dtuxx(t)²₂≤M1+M2uxx(t)⁴₂, (3.23) whereM1andM2are independent oft. We sety(t)= uxx(t)²2,h(t)=M1, and g(t)=M2u_xx(t)²2. For fixedr >0, we use (3.15) to deduce that

_t+r

t g(s)ds≤α1,

_t+r

t h(s)ds≤α2,

_t+r

t y(s)ds≤α3, (3.24) for allt≥t0(Ꮾ, ρ₂), whereα1=M2α3,α2=M1r, andα3=(r/αc_∗)f²2+ρ₂²/α.

Applying uniform Gronwall’s lemma (Lemma 1.3) to the differential inequality (3.23), we conclude that

uxx(t)²₂≤ α3

r +α2

expα1

:=ρ²₃, ∀t≥t0

Ꮾ, ρ2

+r (3.25)

and the proof is complete.

Lemma3.4. Let Hypotheses1.1and3.1be fulfilled,u0∈Ꮾ, and f ∈H. Assume also that (3.1) is satisfied. Then, there exists a constantρ4 independent of tand t2>0, such that

uxxx(t)₂≤ρ4, fort≥t2. (3.26)

Proof. We multiply (1.1a) by−u⁽⁶⁾and integrate overΩto get the equation 1

2 d

dtu_xxx²₂+

Ωa(u)u²_xxxxdx+

ΩA1(u)u²_xu_xxu_xxxxdx + 2

ΩA2(u)u_xu_xxxu_xxxxdx+

ΩA3(u)u²_xxu_xxxxdx +λ

Ω

σ(u)u²_x+σ(u)uxx

uxxxxdx+

Ωb(u)u⁴_xuxxxxdx

= −

Ωfxxuxxxxdx,

(3.27)

whereA1(u)=a(u) + 5b(u),A2(u)=a(u) +b(u), andA3(u)=a(u) + 2b(u).

Similarly toLemma 3.3, we arrive at the inequality d

dtuxxx(t)²₂+c_∗uxxxx(t)²₂≤M3+M4uxxx(t)⁴₂, (3.28) whereM3(ρ1, ρ₂, ρ3) andM4(ρ1, ρ₂, ρ3) are independent oft. Moreover, from in- equality (3.22) we obtain that for fixedr>0

_t+r

t

uxxx(s)²ds≤M1r c_∗ +ρ²3

c_∗

M2ρ²₃r+ 1. (3.29)

Settingy(t)=u_xxx(t)²2,h(t)=M3, andg(t)=M4u_xxx(t)²2, inequality (3.29) implies the following estimates:

_t+r

t g(s)ds≤β1,

_t+r

t h(s)ds≤β2,

_t+r

t y(s)ds≤β3, (3.30) where

β1=M4β3, β2=M3r, β3=M1r c_∗ +ρ²₃

c_∗

M2ρ₃²r+ 1. (3.31)

ApplyingLemma 1.3to the differential inequality (3.28), we conclude that uxxx(t)²₂≤

β3

r +β2

expβ1

:=ρ₄², fort≥t1+r (3.32)

to complete the proof.

Next we discuss certain regularity questions of the solution and the solution operator for problem (1.1).