metaparabolic equations

In this paper, we study the initial boundary value problem for a class of metaparabolic equations. We establish the existence of solutions by the energy techniques. Some results on the regularity, blow-up and existence of global attractor are obtained.


Introduction
In this paper, we study a metaparabolic equation in a bounded domain Ω ⊂ R n (n ≤ 3) with smooth boundary, where γ > 0 is the interfacial energy parameter, k > 0 is the viscosity coefficient, ϕ(∇u) is an intrinsic chemical potential with typical example as where γ 1 is a constant.The term ∆ 2 u denotes the capillarity-driven surface diffusion, and div(ϕ(∇u)) denotes the upward hopping of atoms.
The equation (1.1) is a typical higher order equation, which has an extensive physical background and a rich theoretical connotation.A. Novick-Cohen [10] derived the following equation to study the dynamics of viscous first order phase transitions in cooling binary solutions such as alloys, glasses and polymer mixtures ∂u ∂t = ∆ µ + k ∂∆u ∂t , where u(x, t) is a concentration, µ is the intrinsic chemical potential.If we take µ = ϕ(u) − γ∆u, we obtain the viscous Cahn-Hilliard equation Many authors have paid much attention to the viscous Cahn-Hilliard equation, among which some numerical approaches and basic existence results have been developed [1,3,9,12].The well-known Cahn-Hilliard equation is obtained by setting k = 0, which has been well studied; see for example [2,4,15].In fact, when the influence of many factors, such as the molecular and ion effects, are considered, one has the nonlinear relation div(ϕ(∇u)) in stead of ∆ϕ(u) in right-hand side of above equation, so we obtain the equation (1.1).
On the basis of physical consideration, the equation (1.1) is supplemented by the zero mass flux boundary condition and initial value condition During the past years, many authors have paid much attention to the equation (1.1) for a special case with k = 0 [11,14].B. B. King, O. Stein and M. Winkler [5] studied the equation (1.1) for a special case with k = 0, namely, He proved the existence of solutions for one dimension [6] and two dimensions [7,8].
The purpose of the present paper is devoted to the investigation of properties of solutions with γ 1 not restricted to be positive.We first discuss the regularity.We show that the solutions might not be classical globally.In other words, in some cases, the solutions exist globally, while in some other cases, such solutions blow up at a finite time.The main difficulties for treating the problem (1.1)-(1.3)are caused by the nonlinearity of div(ϕ(∇u)) and the lack of maximum principle.To prove the existence of solutions, the method we use is based on the energy estimates and the Schauder type a priori estimates.In order to prove the blow-up result, we construct a new Lyapunov functional.

Global existence
Let Ω = (0, 1) and consider the following initial-boundary value problem where D = ∂ ∂x , ϕ(z) = −z + γ 1 z 3 with γ, γ 1 and k being constants with γ, k > 0. From the classical approach, it is not difficult to conclude that the problem admits a unique classical solution local in time.So, it is sufficient to make a priori estimates.
Theorem 2.1 If γ 1 > 0, then for any initial data u 0 ∈ H 2 (Ω) with Du 0 (0) = Du 0 (1) = 0 and T > 0, the problem (2.1)-( 2.3) admits one and only one solution Proof.Multiplying both sides of the equation by u and then integrating resulting relation with respect to x over (0, 1), we have ( The Gronwall inequality implies that sup and By the Sobolev imbedding theorem, it follows from (2.5), (2.6), (2.9) that sup Again multiplying both sides of the equation (2.1) by D 4 u and integrating the resulting relation with respect to x over (0, 1), we have 12) The a priori estimates (2.8), (2.9) and (2.12) complete the proof of global existence. (2.16) By the Nirenberg inequality, we have

.18)
Applying Young's inequality to the right-hand side of We have and Here, we have used (2.14), (2.17) and (2.18).Hence, we obtain This completes the proof of global existence.

Regularity
Now, we turn our discussion to the regularity of solutions.
Proof.Set L = (I − kD 2 ) −1 , and Lg = w, namely, w satisfies It is easily seen that Now, we rewrite the equation (2.1) into the form Using the operator L for the above equation, we have Integrating the equation (3.1) with respect to x over (y, y + (∆t) 1/4 ) × (t 1 , t 2 ), where 0 < t 1 < t 2 < T , ∆t = t 2 − t 1 , we see that Integrating the above equality with respect to y over (x, x + (∆t) 1/4 ), we get x N (s, y)dyds.
Here, we have used the mean value theorem, where x * = y * + θ * (∆t) Integrating the above equation with respect to x over (y, y + (∆t) 3/4 ), using the mean value theorem, and (3.3) we have By (2.6) and (2.9), we get Again using (3.5), (3.6) we have Define the linear spaces and the associated operator T : X → X, u → w, where w is determined by the following linear problem

The case of small initial data
In §2, we have proved the global existence of solution of the problem (1.1)-(1.3)for γ 1 > 0. We turn now to the proof of global existence for γ 1 < 0. Without loss of generality, we assume that (2.13) holds.The equation (2.1) may be rewritten as For any fixed t > 0 we define Our goal is to show that N (t) is bounded above.Firstly, multiplying (4.1) by u and integrating with respect to x, we have Integrating the (4.3) over (0, t), we have Next, multiplying (4.1) by u t and integrating with respect to x over (0, 1), and integrating by parts, we have By (4.4), (4.5) we obtain and using the Sobolev inequality and the Poincaré inequality, we obtain Taking (4.6) and (4.8) together yields We conclude that there is a constant C 10 such that EJQTDE, 2010 No. 40, p. 9 provided that u 0 2 is sufficiently small.To show this, we set Assume that u 0 2 is small enough, such that Then we have the assertion In fact, if (4.12) were not sure, then there would exist a t 0 > 0, such that M (t 0 ) = 1 2 and M (t) < 1 2 for t ∈ (0, t 0 ).By (4.9) we obtain Using the second inequality in (4.11), we have The contradiction shows that (4.12) and hence (4.10) holds.Finally multiplying (4.1) by −D 2 u and D 4 u yield the following inequalities As did in section 2, we may easily show that the global solution u satisfies u t , D 4 u, D 2 u t ∈ L 2 (Q T ).The proof is complete.

Blow-up
In the previous sections, we have seen that the solution of the problem (1.1)-(1.3) is globally classical, provided that γ 1 > 0 or γ 1 < 0 and u 0 2 sufficiently small.The following theorem shows that the solution of the problem (1.1)-(1.3)blows up at a finite time for γ 1 < 0 and F (0) ≤ 0. where Now multiplying (1.1) by u and integrating with respect to x, we obtain ) and Poincaré inequality Again −F (0) ≥ 0, hence By u 0 ≡ 0, it follows that u must blow up in a finite time T .

Global attractor
In this section, we are going to prove the existence of attractor as γ 1 > 0. By Theorem 6.To study the existence of a global attractor, we have to find a closed metric space and prove that there exists a global attractor in the closed metric space.Since the total mass is conserved for all time, it is not possible for us to have a global attractor for the whole space without any constraints.Instead, we consider a series of subspaces with constraints as follow Ω udx ≤ κ, for any given positive constant κ.
We let where κ > 0 is a constant.It is easy to see that the restriction of {S(t)} on the affined space X κ is a well defined semigroup.
Theorem 6.1 For every κ chosen as above, the semiflow associated with the solution u of the problem (1.1)-( 1.3) possesses in X κ a global attractor A κ which attracts all the bounded set in X κ .
In order to prove Theorem 6.1, here we establish some a priori estimates for the solution u of problem (1.1)- (1.3).In what follows, we always assume that {S(t)} t≥0 is the semigroup generated by the weak solutions of problem (1.1)-(1.3)with initial data u 0 ∈ H 2 (Ω).Lemma 6.1 There exists a bounded set B κ whose size depends only on κ and Ω, in X κ such that for all the orbits starting from any bounded set B in X κ , ∃ t 1 = t 1 (B) ≥ 0 such that ∀ t ≥ t 1 all the orbits will stay in B κ .
Proof.By Young inequality, it follows from (2.4) that 1 2 By Poincaré inequality, we have Using Friedrichs inequality, we get Du 2 ≤ D 2 u 2 .So, if γ sufficiently large, we have Thus for initial in any bounded set B ⊂ X κ , there is a uniform time t 0 (B) depending on B such that for t ≥ t 0 (B), The lemma is proved.Lemma 6.2For any initial data u 0 in any bounded set B ⊂ X κ , there is a t 2 (B) > 0 such that u(t) H 3 ≤ C, ∀ t ≥ t 2 > 0, which turns out that t≥t2 u(t) is relatively compact in X κ .
Proof.Firstly, integrating (6.2) over (t, t + 1), we have Combining (2.11) with (6.4) and using the uniform Gronwall inequality, we have that there is a uniform time t 2 (B) depending on B such that for t ≥ t 2 (B), The lemma is proved.
Then by Theorem I.1.1 in [13], we immediately conclude that A κ = ω(B κ ), the ω-limit set of absorbing set B κ is a global attractor in X κ .By Lemma 6.2, this global attractor is a bounded set in H 3 .Thus the proof of Theorem 6.1 is complete.

Theorem 4 . 1
If γ > 3 2π 2 , and u 0 2 is sufficiently small, then there exists a unique global solution u with ut , D 4 u, D 2 u t ∈ L 2 (Q T ) to (2.1)-(2.3).Proof.As mentioned before, it needs only to obtain a priori estimates for smooth solution u.In what follows C j (j = 1, 2 • • •) denote the constants independent of u and t.Set f = D(γ 1 (Du) 3 ).

1 )(D 3 u) 2 dx + k 1 0D 2
Similar to the above, multiplying both sides of the equation byD 2 u, u t D 2 udx ≤ C. (6.2)Hence there is a uniform time t 1 (B) depending on B such that for t ≥ t 1 (B),
1, we can define the operator semigroup {S(t)} t≥0 in H 2 space as