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Abstract and Applied Analysis Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation
Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation
Karachalios, Nikos, Stavrakakis, Nikos, Xanthopoulos, Pavlosこの本はいかがでしたか？
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巻:
2003
年:
2003
言語:
english
ジャーナル:
Abstract and Applied Analysis
DOI:
10.1155/s1085337503210022
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS 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 attractor for the corresponding dynamical system. 1. Introduction We consider the following nonlinear parabolic initial boundary value problem in the open bounded interval Ω ⊂ R ut − a(u)uxx − b(u)u2x − λσ(u) = f (x), x ∈ Ω, t > 0, u(x,0) = u0 (x), u∂Ω = 0, t > 0. (1.1a) (1.1b) (1.1c) This problem extends the well studied porous medium diﬀusion, since no certain relationship between the coeﬃcients a(u) and b(u) is assumed. Let us mention that special cases of this system may typically arise in plasma physics within the context of the fluid treatment of charged particles, and in densitydependent reaction diﬀusion processes in mathematical biology. Naturally enough, these systems imply only positive values for u(x,t); however, in the following treatment, we do not impose such a restriction. In order to demonstrate a specific casemodelled system, we consider the collisionless evolution equation for the electron pressure P = nT, which, if we ignore viscosity, gets the following form in the xdirection (see, Balescu [5]) 3 3 5 Pt = −qx − uPx − Pux , 2 2 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 (1.2) 522 Semibounded evolution equation where u represents the electron velocity and q 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 5 3 Pt = −qx + cPx2 + cPPxx . 2 2 2 (1.4) We see that the first term on the righthand side corresponds to porous medium diﬀusion (not con; sidered here), whereas the other two terms constitute a specific case of (1.1a), with a(P) = (5/3)cP and b(P) = c. Concerning the applications in the dynamics of cell populations, with a spatial distribution of cells, we may associate an energy density e(u), that is an internal energy per unit volume of an evolving spatial pattern, where u(x,t) denotes the cell density (see [6, 14]). In this case, the total energy E(u) in a volume V is given by E(u) = V e(u)dx. (1.5) The change in energy δE, that is the work done in changing states by an amount δu, is given by the variational derivative δE/δu which defines a potential µ(u) = δE = e (u). δu (1.6) The gradient of the potential µ produces a flux J, 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 density u is ∂u = a(u)ux x , ∂t 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 holds a (u) = b(u), that is the porous medium case. Also, the nonlinearity σ(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 diffusion). In [4], the existence of a global attractor for the onedimensional porous medium equation, attracting all orbits starting from L∞ initial data, is demonstrated. Extensive studies in [1, 13, 15] show that the ωlimit set is contained Nikos Karachalios et al. 523 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 ∈ C 2 (R), λ ∈ R, and there exists c∗ > 0 such that a(s) ≥ c∗ (i.e., we consider nondegenerate but nonmonotone diﬀusion). Due to the nonmonotonicity, the standard compactness methods on existence 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],H 2 ∩ H01 (Ω)), the space of weakly continuous functions u : [0,T] → H 2 ∩ H01 (Ω). The existence of a global attractor in the phase space H = H 2 ∩ H01 (Ω) is proved in Section 3. The result is shown assuming monotonicity for the nonlinearity b(·), 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 in Section 2. Further, using this result, we may define the semigroup S(t) : u0 ∈ H → u(t) ∈ H, corresponding to our problem. We conclude by recalling some wellknown results, which will be frequently used (see, [16, 17, 18, 19]). Lemma 1.2 (GagliardoNirenberg inequality). Let 1 ≤ p, q,r ≤ ∞, j an integer, 0 ≤ j ≤ m, and j/m ≤ θ ≤ 1. Then j D u ≤ const u p 1−θ m θ D u r, q u ∈ Lq ∩ W m,r (Ω), Ω ⊆ Rn , (1.9) where j 1 1−θ 1 m = +θ − + . p n r n q (1.10) If m − j − n/r is not a nonnegative integer, then the inequality holds for j/m ≤ θ <1. Lemma 1.3 (uniform Gronwall). Let g,h, y be three positive locally integrable functions for t0 ≤ t < ∞ which satisfy t+r t dy ≤ g y + h, dt g(s)ds ≤ α1 , t+r t ∀t ≥ t0 , h(s)ds ≤ α2 , t+r t (1.11) y(s)ds ≤ α3 , 524 Semibounded evolution equation for all t ≥ t0 , where α1 , α2 , and α3 are positive constants. Then α y(t + r) ≤ 3 + α2 exp α1 , r ∀t ≥ t0 . (1.12) We also use the short (equivalent) norms ux 2 , uxx 2 , and uxxx 2 in H01 (Ω), 2 H ∩ H01 (Ω), and H 3 ∩ H01 (Ω), respectively (see Section 3). From the embedding H k ∩ H01 (Ω) Cbk−1 (Ω), k = 1,2,..., and the Poincaré inequality (see [9, page 242]), we have (k−1) u ≤ const u ∞ H k ∩H01 ≤ const u(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 an admissible triple of Banach spaces, which is defined as follows (see [17, 18] and [19, page 784]). Definition 2.1. An admissible triple V H W has the following properties: (i) H is a real separable Hilbert space with scalar product (··)H , (ii) {V,W } is a dual pair of real separable Banach spaces with the corresponding bilinear form ·, · (i.e., ·, · is continuous, w,v = 0, for every w ∈ W, implies v = 0, and w,v = 0, for every v ∈ V , implies w = 0), (iii) the embeddings V H W are continuous and dense, (iv) it holds h,v = (hv)H , for all h ∈ H, v ∈ V . Clearly, an admissible triple generalizes the notion of the evolution triple, in the sense that for an admissible triple it may hold W = 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 = H 4 ∩ H01 (Ω), H = H 2 ∩ H01 (Ω), W = L2 (Ω). (2.1) Lemma 2.2. The embedding V H W for 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 = Ω vw + wvxxxx dx, ∀v ∈ V, w ∈ W. (2.2) Now, it is easy to check that the inner product stemming from the bilinear form ·, · (wv)H = Ω vw + wxx vxx dx, for every w ∈ H, v ∈ H (2.3) Nikos Karachalios et al. 525 induces an equivalent norm in H. We also have that w,v = vw + wvxxxx dx Ω ≤ w 2 ≤c w v W w 2 vxxxx 2 2+ v (2.4) V, hence the bilinear form ·, · is continuous. Now assume that, for some w ∈ W, it holds w,v = 0, for every v ∈ 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, For this solution v, we have that v ∈ V. (2.5) w2 dx, (2.6) 0 = w,v = Ω which implies that w = 0 and the proof is complete. We introduce the nonlinear operators A,B : V → W defined by Bu = −b(u)u2x . Au = −a(u)uxx , (2.7) The following results outline the basic properties of the operators A and B. Proposition 2.3. The operator A + B : H → W is bounded on bounded sets of H. Proof. Let B = BH (R) be a closed ball in H. We will show that there exist constants K1 (R) and K2 (R) such that Au 2 ≤ K1 (R) u H, Bu 2 ≤ K2 (R) u H, ∀u ∈ B. (2.8) Since a, b, σ ∈ C 2 (R) and the embedding H Cb1 (Ω) is continuous, it follows that there exist constants C1,m (R) and C2,m (R), m = 0,1,2, such that sup a(m) u(x) ≤ C1,m (R), m = 0,1,2, (2.9) sup b(m) u(x) ≤ C2,m (R), m = 0,1,2. (2.10) x ∈Ω x ∈Ω Using (2.9), (2.10), and the fact that H01 (Ω) is a generalized Banach algebra, we may obtain the inequalities Au 2 ≤ sup a u(x) uxx 2 ≤ K1 (R) u Bu 2 ≤ sup b u(x) u2x 2 ≤ constsup b u(x) u x ∈Ω x ∈Ω ≤ K2 (R) u x ∈Ω H. H, 2 H (2.11) 526 Semibounded evolution equation Finally, we conclude that (A + B)u ≤ K(R) u 2 H, (2.12) where K(R) = max{K1 (R),K2 (R)}. Proposition 2.4. The operator A + B : H → W is locally Lipschitz continuous. Proof. Let u,v ∈ B = BH (R) be a closed ball in H. We have that Au − Av 2 ≤ a(u) − a(v) vxx 2 + a(u) uxx − vxx 2 . (2.13) From the mean value theorem and (2.9), we get a u(x) − a v(x) ≤ C1,1 (R)u(x) − v(x), a u(x) − a v(x) ≤ C1,2 (R)u(x) − v(x). (2.14) (2.15) Therefore, a(u) − a(v) vxx 2 ≤ C1,1 (R)2 u − v 2 vxx 2 ≤ C(R) u − v ∞ 2 2 a(u) uxx − vxx 2 ≤ C 2 (R)uxx − vxx 2 ≤ C(R) u − v 2 , H 1,0 2 2 2 H, (2.16) where C(R) is a common symbol for the constants. Similar inequalities hold for the operator B. So finally it holds that (A + B)u − (A + B)v ≤ C(R) u − v 2 H. (2.17) Proposition 2.5. The operator A + B : H → W is semibounded. Proof. By definition, it must be proved that there exists a monotone increasing function d1 ∈ C 1 (R) such that (A + B)u,u ≥ −d1 u 2 H for every u ∈ V. , (2.18) Let u ∈ C0∞ (Ω) ∩ C(Ω). For the operator A, it holds Au,u = Ω Auudx + Ω Auuxxxx dx. (2.19) Nikos Karachalios et al. 527 Integration by parts in the second integral on the righthand side of (2.19) gives − Ω a(u)uxx uxxxx dx = − 1 2 Ω + Ω a (u)u2x u2xx dx − 1 2 Ω a (u)u3xx dx (2.20) a(u)u2xxx dx. Using Lemma 1.2, we obtain the inequality uxx ≤ const u 4 3/4 1/4 uxxx 2 , 2 (2.21) which, with the aid of (2.9) and Young’s inequality, gives the following estimate: − 1 2 Ω a (u)u2x u2xx dx − 1 2 ≥ −C1,2 ux Ω 2 a (u)u3xx dx 2 uxx − C1,1 uxx uxx 2 2 2 4 3/2 1/2 4 ˆ ˆ ≥ −C1 u H − C2 u H u 2 uxxx 2 3/2 ≥ −Cˆ1 u 4H − Cˆ3 u 3/2 H uxxx 2 2 c∗ ≥ −Cˆ1 u 4H − Cˆ4 u 6H − uxxx 2 . ∞ (2.22) 2 For the first integral of the righthand side of (2.19), we have − Ω a(u)uxx udx ≥ −C1,0 u uxx ∞ 1 ≥ −Cˆ0 u 2 H. (2.23) Using Hypothesis 1.1, (2.19), (2.20), (2.22), (2.23), and density arguments, we obtain that Au,u ≥ −Cˆ0 u 2 H − Cˆ1 u 4 H − Cˆ4 u 6 H := −d1,1 u 2 H . (2.24) A similar procedure may be followed for the operator B, to derive the relation Bu,u ≥ −d1,2 u 2 H . (2.25) Finally, from estimates (2.24) and (2.25) we get that there exists a monotone increasing C 1  function d1 : R → R satisfying (2.18). The previous propositions enable us to show local existence of solutions. The result is stated as follows. 528 Semibounded evolution equation Theorem 2.6. Let u0 , f ∈ H. Assume that Hypothesis 1.1 is satisfied. Then there exists T > 0 such that problem (1.1) has a unique solution u ∈ Cw [0,T],H , ut ∈ Cw [0,T],W . (2.26) Moreover, the solution u : [0,T] → W is Lipschitz continuous. Proof. (A) Existence: the first step is to show existence of at least one solution in a finite dimensional subspace Vn = span{e1 ,...,en } of V , where {ei }i≥1 is an orthonormal basis of Vn with respect to (··)H . It holds that n Vn = V H. We define the linear and continuous operator P̃n : W → V as n P̃n w = w ∈ W. w,ei ei , (2.27) i Now, the Galerkin equation for problem (1.1) on Vn V H reads un (t) + P̃n (A + B)un (t) = P̃n Cun (t), t ∈ [0,T], un (0) = P̃n u0 , (2.28) where Cun (t) = λσ un (t) + f . (2.29) Using Propositions 2.3 and 2.4, Peano’s theorem justifies the existence of a C 1 solution for (2.28), un : [0,T0 ] → Vn , for some T0 > 0 which depends on n. The next step is to obtain an a priori estimate for un in H. Note that P̃n : H → Vn is an orthogonal projection onto the space Vn , since it holds P̃n u = n i (uei )H ei , u ∈ H. Since un is continuous on [0, T0 ], (2.28) implies that un un H = − P̃n (A + B)un un H + P̃n Cun 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 ∈ C 1 (R) such that Cu,u ≤ d2 u 2 H , ∀u ∈ V. (2.31) Hence, from (2.18), (2.30), and (2.31) we obtain the diﬀerential inequality d un (t)2 ≤ 2d un (t)2 , H H dt t ∈ 0,T0 , (2.32) Nikos Karachalios et al. 529 where un (0) H = Pn u0 H ≤ u0 H . Since the function d(·) is Lipschitz continuous as a C 1 function, we may apply the theorem of PicardLindelöf to conclude that there exists a T > 0, this time independent of n, such that un (t)2 ≤ max g(t) ≤ R, H t ∈[0,T] t ∈ [0,T]. (2.33) Finally, using standard continuation arguments, we can extend the solution un to the interval [0,T]. Now, from (2.33) we have that there exists a subsequence, denoted again by {un }, such that un (t) in H, as n −→ ∞, u(t), (2.34) at least in a dense countable subset of [0,T]. Let v ∈ Vk H, k ≤ n. Since P̃n v = v, for every k ≤ n, it follows that un (t)v H = − P̃n (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 interval [0,T]. Finally, passing to the limit to (2.35) and using density of k Vk in H, we obtain that u ∈ Cw ([0,T],H), ut ∈ Cw ([0,T],W) is a solution for problem (1.1) and as a consequence, u : [0,T] → W is Lipschitz continuous. (B) Uniqueness: the diﬀerence of solutions w = u − v of 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))vx2 + b(u)(u2x − vx2 ), and Σ(u, v) = σ(u) − σ(v). Multiplying (2.36) by u and integrating over Ω, we obtain the equation 1 d w 2 dt 2 2+ + Ω Ω − + a (u)vx wwx dx + a (u)ux wwx dx + Ω Ω b(u) − b(v) Ω Ω a(u) − a(v) wx vx dx vx2 w dx − a(u)wx2 dx − λ Ω a (u) − a (v) vx2 w dx Ω b(u) u2x − vx2 w dx σ(u) − σ(v) w dx = 0. (2.37) 530 Semibounded evolution equation Using estimate (2.33) and relations (2.9), (2.10), and (2.15) the following estimates are derived: 2 2 a (u) − a (v) vx2 w dx ≤ C1,2 vx ∞ w 2 ≤ C(R) w Ω a (u)vx wwx dx ≤ C1,1 vx w 2 wx ∞ 2 Ω 2 ≤ 0 wx 2 + C(R) w 22 . 2 2, (2.38) The rest of the integrals in (2.37) can be estimated in a similar way. Hence, for suﬃciently small 0 , we get the inequality 1 d w(t)2 + c∗ wx 2 ≤ C w(t)2 . W 2 W 2 dt 2 Application of the standard Gronwall’s lemma implies uniqueness. (2.39) 3. Existence of a global attractor in H 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 functions b, σ satisfy the following hypothesis. Hypothesis 3.1. b (s) ≤ 0 and there exist cm > 0, such that σ (m) (s) ≤ cm s, for all m = 0,1,2. First, we prove that under the extra hypothesis, Hypothesis 3.1, the unique local solution u(x,t) of problem (1.1), obtained in Theorem 2.6, exists globally in time. We denote by λ∗ the positive constant induced by Poincaré’s inequality. Lemma 3.2. Let Hypotheses 1.1 and 3.1 be fulfilled and u0 , f ∈ H. Assume also that λ< c∗ λ∗ . 2c0 (3.1) Then there exists a constant ρ2 independent of t, such that, limsup ux (t)2 ≤ ρ2 . (3.2) t →∞ Proof. We multiply (1.1a) by −uxx and integrate over Ω to get 1 d u x 2 + 2 2 dt +λ Ω a(u)u2xx dx + Ω Ω b(u)u2x uxx dx σ(u)uxx dx = (3.3) Ω f uxx dx. Nikos Karachalios et al. 531 Using Hypothesis 1.1, we observe that Ω 2 a(u)u2xx dx ≥ c∗ uxx 2 , (3.4) whereas from Hypothesis 3.1 we have Ω b(u)u2x uxx dx = − 1 3 Ω b (u)u4x dx ≥ 0. (3.5) Furthermore, Hypothesis 3.1, together with Poincaré’s inequality 1/2 ≤ λ− ux 2 , ∗ (3.6) 2 −1 σ(u)uxx dx ≤ λc0 u 2 uxx 2 ≤ λλ∗ c0 uxx 2 . (3.7) u 2 implies that λ Ω Relations (3.3), (3.4), and (3.7) imply that d ux (t)2 + αuxx (t)2 ≤ 1 f 2 2 dt c∗ 2 2, (3.8) where α = c∗ − 2c0 λλ−∗1 . Applying again Poincaré’s inequality (3.6) to the above estimate (3.8), we get d ux (t)2 + αλ∗ ux (t)2 ≤ 1 f 2 2 dt c∗ 2 2. (3.9) If assumption (3.1) is satisfied, that is, α > 0 Gronwall’s lemma leads to the following estimate: ux (t)2 ≤ ux (0)2 exp − αλ∗ t + 2 2 1 f αc∗ λ∗ 2 2 1 − exp − αλ∗ t . (3.10) Letting t → ∞, from estimate (3.10) we obtain that 2 limsup ux (t)2 ≤ ρ22 , t →∞ where ρ22 = (1/αc∗ λ∗ ) f 2 2 and the proof is completed. (3.11) 532 Semibounded evolution equation Let Ꮾ be a bounded set of H, included in a ball BH (0,M) of H, centered at 0 of radius M. Assuming that u0 ∈ Ꮾ, we infer from Lemma 3.2 that for ρ2 > ρ2 , there exists t0 (Ꮾ,ρ2 ) > 0 such that for t ≥ t0 (Ꮾ,ρ2 ) ux (t) ≤ ρ , 2 2 u(t) ≤ ρ1 = λ−1/2 ρ . ∗ 2 2 (3.12) Integrating (3.8) with respect to t, it follows that for every r > 0 t+r α uxx (s)2 ds ≤ r 2 t 2 2 2 + ux (t) 2 . f c∗ (3.13) Once again, letting t → ∞, we obtain from inequality (3.12) that t+r limsup t →∞ uxx (s)2 ds ≤ r 2 t αc∗ ρ22 , α for every r > 0. (3.14) ρ22 , α for every r > 0. (3.15) 2 2+ f and for t ≥ t0 (Ꮾ,ρ2 ) t+r t uxx (s)2 ds ≤ r 2 αc∗ f 2 2+ Lemma 3.3. Let Hypotheses 1.1 and 3.1 be fulfilled, u0 ∈ Ꮾ, and f ∈ H. Assume also that (3.1) is satisfied. Then there exists a constant ρ3 independent of t, and t1 > 0 such that uxx (t) ≤ ρ3 , 2 for t ≥ t1 . (3.16) Proof. Multiply (1.1a) by uxxxx and integrate over Ω to get 1 d uxx 2 + 2 2 dt Ω a (u)ux uxx uxxx dx + +2 + =− Ω Ω Ω a(u)u2xxx dx Ω b b(u)ux uxx uxxx dx + λ Ω σ (u)ux uxxx dx (3.17) (u)u3x uxxx dx fx uxxx dx. Using inequalities (1.13), (3.12), and Hypothesis 1.1, we obtain that inequalities (2.9) and (2.10) hold, for all t ≥ t0 (Ꮾ,ρ2 ), with R replaced by ρ2 . It follows that a (u)ux uxx uxxx dx ≤ C1,1 ux uxx uxxx ∞ 2 2 Ω 2 ≤ C1,1 const uxx 2 uxxx 2 4 2 ≤ C1 uxx 2 + 1 uxxx 2 . (3.18) Nikos Karachalios et al. 533 Applying Lemma 1.2, we obtain the inequality ux ≤ const u 6 2/3 1/3 uxx 2 , 2 (3.19) which can be used to get the estimate b (u)u3 uxxx dx ≤ C2,1 ux 3 uxxx x 6 2 Ω 2 ≤ C2,1 const u 2 uxx 2 uxxx 2 4 2 ≤ C2 uxx + 1 uxxx 2 . (3.20) We also have that the estimate λ Ω σ (u)ux uxxx dx ≤ λc1 u ∞ ux uxxx 2 2 2 ≤ λc1 const uxx 2 uxxx 2 4 2 ≤ C3 uxx + 1 uxxx . 2 (3.21) 2 The rest of the integral terms in (3.17) can be bounded similarly. Thus, for suﬃciently small 1 , we get the inequalities d uxx (t)2 + c∗ uxxx (t)2 ≤ M1 + M2 uxx (t)4 , 2 2 2 dt 2 4 d uxx (t) ≤ M1 + M2 uxx (t) , 2 2 dt (3.22) (3.23) where M1 and M2 are independent of t. We set y(t) = uxx (t) 22 , h(t) = M1 , and g(t) = M2 uxx (t) 22 . For fixed r > 0, we use (3.15) to deduce that t+r t t+r g(s)ds ≤ α1 , t h(s)ds ≤ α2 , t+r t y(s)ds ≤ α3 , (3.24) for all t ≥ t0 (Ꮾ,ρ2 ), where α1 = M2 α3 , α2 = M1 r, and α3 = (r/αc∗ ) f 22 + ρ22 /α. Applying uniform Gronwall’s lemma (Lemma 1.3) to the diﬀerential inequality (3.23), we conclude that uxx (t)2 ≤ 2 α3 + α2 exp α1 := ρ32 , r ∀t ≥ t0 Ꮾ,ρ2 + r (3.25) and the proof is complete. Lemma 3.4. Let Hypotheses 1.1 and 3.1 be fulfilled, u0 ∈ Ꮾ, and f ∈ H. Assume also that (3.1) is satisfied. Then, there exists a constant ρ4 independent of t and t2 > 0, such that uxxx (t) ≤ ρ4 , 2 for t ≥ t2 . (3.26) 534 Semibounded evolution equation Proof. We multiply (1.1a) by −u(6) and integrate over Ω to get the equation 1 d uxxx 2 + 2 2 dt a(u)u2xxxx dx + Ω +2 Ω +λ =− Ω A1 (u)u2x uxx uxxxx dx A2 (u)ux uxxx uxxxx dx + Ω Ω σ (u)u2x + σ (u)uxx Ω A3 (u)u2xx uxxxx dx uxxxx dx + (3.27) Ω b (u)u4x uxxxx dx fxx uxxxx dx, where A1 (u) = a (u) + 5b (u), A2 (u) = a (u) + b(u), and A3 (u) = a (u) + 2b(u). Similarly to Lemma 3.3, we arrive at the inequality d uxxx (t)2 + c∗ uxxxx (t)2 ≤ M3 + M4 uxxx (t)4 , 2 2 2 dt (3.28) where M3 (ρ1 ,ρ2 ,ρ3 ) and M4 (ρ1 ,ρ2 ,ρ3 ) are independent of t. Moreover, from inequality (3.22) we obtain that for fixed r > 0 t+r 2 uxxx (s)2 ds ≤ M1 r + ρ3 M2 ρ2 r + 1 . 3 c∗ t c∗ (3.29) Setting y(t) = uxxx (t) 22 , h(t) = M3 , and g(t)=M4 uxxx (t) 22 , inequality (3.29) implies the following estimates: t+r t t+r g(s)ds ≤ β1 , t h(s)ds ≤ β2 , t+r y(s)ds ≤ β3 , (3.30) M1 r ρ32 + M2 ρ32 r + 1 . c∗ c∗ (3.31) t where β 2 = M3 r , β 1 = M4 β 3 , β3 = Applying Lemma 1.3 to the diﬀerential inequality (3.28), we conclude that uxxx (t)2 ≤ 2 to complete the proof. β3 + β2 exp β1 := ρ42 , r for t ≥ t1 + r (3.32) Next we discuss certain regularity questions of the solution and the solution operator for problem (1.1). Nikos Karachalios et al. 535 Proposition 3.5. Let Hypotheses 1.1 and 3.1 be fulfilled and u0 , f ∈ H. Then, for the unique solution of (1.1), it holds that u ∈ C(0,T;H), for every T > 0. Moreover, the mapping S(t) : u0 ∈ H → u(t) ∈ H is continuous. Proof. We will divide the proof to two parts. (A) Continuity of Solutions. Consider the dense embeddings V H V ∗. (3.33) A consequence of relation (3.28) is that u ∈ L2 (0,T;V ), for every T > 0. Also, it can be easily proved that ut ∈ L2 (0,T;W). Taking into account the continuous embedding L2 (0,T;W) L2 (0,T;V ∗ ), it follows that u ∈ ᐃ ≡ u ∈ L2 (0,T;V ), ut ∈ L2 (0,T;V ∗ ) C(0,T;H). (3.34) (B) Continuity of the solution mapping. Multiply (2.36) by wxxxx and integrate over Ω to get the following relation: 1 d wxx 2 + 2 2 dt 2 a(u)wxxx dx + Ω + b (u)vx2 wx wxxx dx + Ω + Ω Ω a (u)wx vxx wxxx dx Ω b (u) − b (v) vx3 wxxx dx Ω Ω b (u) ux + vx ux wx wxxx dx b(u) ux + vx wxx wxxx dx + +2 a(u) − a(v) vxxx wxxx dx + Ω + a (u)ux wxx wxxx dx a (u) − a (v) vx vxx wxxx dx + Ω + Ω Ω b(u) uxx + vxx wx wxxx dx b(u) − b(v) vx vxx wxxx dx + λ Ω σ(u) − σ(v) wxxxx dx = 0. (3.35) The integral terms in the equation above, may be estimated as follows: a(u) − a(v) vxxx wxxx dx ≤ C1,1 w ∞ vxxx 2 wxxx 2 Ω ≤ K1 wxx 2 vxxx 2 wxxx 2 2 2 2 ≤ K2 vxxx 2 wxx 2 + 2 wxxx 2 , b(u) uxx + vxx wx wxxx dx ≤ C2,0 wx uxx + vxx wxxx ∞ 2 2 Ω ≤ K3 wxx 2 uxx + vxx 2 wxxx 2 2 2 ≤ K4 wxx 2 + 2 wxxx 2 . (3.36) 536 Semibounded evolution equation The inequality obtained by this procedure, for suﬃciently small 2 , is d wxx (t)2 + c∗ wxxx (t)2 ≤ M0 (t)wxx (t)2 , 2 2 2 dt 2 M0 (t) = C1 + C2 vxxx (t)2 . (3.37a) (3.37b) Since the solution v ∈ L2 (0,T;H 3 ∩ H01 (Ω)) (e.g., see Lemma 3.4), the function M0 (t) is integrable on the interval [0,T]. Therefore, the standard Gronwall lemma is applicable to inequality (3.37a) to obtain wxx (t)2 ≤ C3 wxx (0)2 , 2 2 C3 = exp max M0 (t) . t ∈[0,T] (3.38) Inequality (3.38) implies the continuity of the mapping S(t) : u0 ∈ H → u(t) ∈ H. Now, we are allowed to define a dynamical system in H as the mapping S(t) : u0 ∈ H −→ u(t) ∈ H (3.39) associated to problem (1.1). We conclude with the following result. Theorem 3.6. If f ∈ H, then the semigroup S(t) possesses global attractor Ꮽ in H. Proof. Restating the result of Lemma 3.3 and taking into account inequality (3.25) for some fixed r > 0, we have that the closed ball in H, Ꮾ1 = φ ∈ H : φ H ≤ ρ3 , (3.40) is a bounded absorbing set for the semigroup S(t), that is, for every bounded set Ꮾ in H, there exists t1 (B) > 0, such that S(t)Ꮾ ⊂ Ꮾ1 , for every t ≥ t1 (Ꮾ). On the other hand, Lemma 3.4 implies that there exists t2 (Ꮾ) > 0 such that S(t)Ꮾ ⊂ B2 for t ≥ t2 (Ꮾ), where Ꮾ2 = φ ∈ X : φ X ≤ ρ4 (3.41) is a closed ball in X := H 3 ∩ H01 (Ω). The set Ꮾ2 is bounded in X and relatively compact in H and the semigroup S(t) is uniformly compact. Hence, the set Ꮽ = ω(Ꮾ) is a compact attractor for the semigroup S(t). 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Nikos Karachalios: Department of Statistics and Actuarial Science, School of Sciences, University of the Aegean, Karlovassi, 83200 Samos, Greece Email address: karan@aegean.gr Nikos Stavrakakis: Department of Mathematics, National Technical University, Zografou Campus, 157 80 Athens, Greece Email address: nikolas@central.ntua.gr Pavlos Xanthopoulos: Department of Mathematics, National Technical University, Zografou Campus, 157 80 Athens, Greece Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014