Abstract and Applied Analysis

Volume 2013 (2013), Article ID 372726, 11 pages

http://dx.doi.org/10.1155/2013/372726

Research Article

Hyperbolic Relaxation of a Fourth Order Evolution Equation

Renato Colucci and Gerardo R. Chacón

Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 No. 43-82, Bogotá, Colombia

Received 26 November 2012; Revised 30 January 2013; Accepted 3 February 2013

Academic Editor: Juan J. Nieto

Copyright © 2013 Renato Colucci and Gerardo R. Chacón. 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.

Abstract.

We propose a hyperbolic relaxation of a fourth order evolution equation, with an inertial term , where . We prove the existence of several absorbing sets having different regularities and the existence of a global attractor that is bounded in .

1. Introduction

Leting be an open interval, with , we consider the following initial-boundary value problem for :

where the function is the so-called double-well potential, and are positive parameters.

Problem (1), with , was proposed in [1] where the global dynamics was studied. In particular, the dynamical behavior of the solutions for small values of the parameter was studied by means of numerical experiments. The existence of three well-differentiated time scales with peculiar dynamical behavior was showen. In the first time scale of order there is the formation of microstructure (see [2]) in the region where the gradient of the initial datum falls in the nonconvex region of ; this phenomenon produces a drastic reduction of the energy of the initial datum. In the second time scale of order the equation exhibits a heat equation-like behavior in the convex regions while slow motion in the nonconvex ones. In the last time scale of order the equation shows a finite-dimensional behavior: the solution is approximately the union of consecutive segments and the dynamic is slow.

In [3], the third time scale was studied; the authors proved the existence of a global attractor (see [4]) that is bounded in . The time for which the solutions enter the absorbing set is of order and it is consistent with the estimates found in [1]. Moreover the authors proved the existence of an exponential attractor with finite fractal dimension of order . In [5] the authors proved the existence of an inertial manifold (see [6]) whose dimension is of order , and by the -dimensional volume elements methods (see [7]) an estimate of the dimension of the global attractor of order was found. This estimate is also consistent with the numerical experiments developed in [1]; in fact it was found that the wave length of the microstructure is of order .

In the last years the viscous and no viscous hyperbolic relaxation of the Cahn-Hilliard equation has been extensively investigated. The model was proposed in [8] while in [9] the existence of a family of exponential attractors was proved. The viscous and nonviscous perturbation has been studied in [10] where the existence of a family of global attractors that are upper semicontinuous with respect to the vanishing of perturbations parameters was proved. These results have been extended in 2 and 3 dimensions; see for example [11, 12] and the references therein.

Due to the similarity of problem (1) to the Cahn-Hilliard equation we consider it interesting to study the hyperbolic relaxation of the fourth order evolution equation proposed in [1]. In particular if is the solution of the Cahn-Hilliard equation

with Neumann boundary conditions:

then

is the solution of (1), with , with the corresponding boundary conditions:

In the present work we put the problem in the correct mathematical framework and prove the existence of a global attractor while we have left the proof of the existence of exponential attractors for a forthcoming paper. In Sections 2 and 3 we define the solution semigroup in the appropriate spaces and present some important energy estimates. In Section 4 we prove the existence of several absorbing sets with different regularities while in the last section we prove the existence of the global attractor.

2. Preliminaries

We begin this section by defining the following Hilbert spaces that will be helpful for our analysis:

where

is the domain of the differential operator . The above spaces are equipped with the norms

where represents the norm. We will denote by the inner product in . We recall that for all with , we have

Throughout the paper we will use two norms that are equivalent to (9) and (10) in order to simplify the computation. Given functions and we define the function on by

where .

Proposition 1. For all the function induces a norm equivalent to the norm on .

Proof. By Schwartz inequality (11) and using the fact that , , we get

From the previous inequality and by definition of we get

By a different application of Schwartz inequality and from (11) we get

Combining (13) and (15) we get

and as a consequence

The proof of the following theorem follows from classical applications of the Faedo-Galerkin method. We will only show a Lipschitz estimate that will be needed for further computations.

Theorem 2 2. For every there exists a unique solution for the initial value problem (1) such that

If, moreover, , then:

Proposition 3 3. For any constant there exists a positive constant such that, for any initial data , with , one has

where is the solution semigroup of the problem (1).

Proof. Let , , two solutions of (1) with initial data and . Let, then we write

We multiply the above equation by in ; then

where we have used the following inequality (see [13] and inequality (11)):

The constant can be explicitly computed using the estimates of the absorbing set presented in the next section.

From the last inequality we get

and then using Gronwall's lemma we get

3. A Priori Estimates

In this section we provide useful a priori estimates of energy type. Equation (1) admits a Liapunov functional of the form:

that is not increasing along the solutions; in fact if we multiply (1) by and integrate over we obtain

Moreover, integrating the previous inequality on with respect to time we get

Then for any fixed initial data we have that the corresponding solution satisfies

Moreover if we integrate over we get the integral control

We consider an important estimate that will be useful later. Let be a parameter to be determined later; if we multiply (1) by we obtain

By summing (27) and (31) we get

Using the expression of the energy we can rewrite the previous in the following way:

We will estimate some of the terms of the previous inequality in the following lemma.

Lemma 4. Fix then for all one has

Proof. By Holder's inequality we get

then we use Poincaré's inequality and the fact that to conclude that

Therefore, for a positive constant to be determined later, we have that

Consequently, by choosing the parameters and we have that

Now, using the previous lemma and (33) we conclude that

If we set

then from (39) we get

Now, integrating we have that

The previous inequality will be used in the next section to show the existence of absorbing sets for the problem (1).

We remark that the following inequality holds for the functions , , and :

4. Absorbing Sets

In this section we will show the existence of several absorbing sets for the solution semigroup of (1) in the space . Also, assuming further regularity of the initial data, the existence of a more regular absorbing sets is also shown.

First, notice the following estimate of the nonlinear term of the energy functional defined in (26). Using (23) we get

We will use this inequality several times in what follows.

4.1. Absorbing Set in

Proposition 5. For all the set

is bounded, absorbing, and positively invariant for the semigroup in .

Proof. The set is positively invariant, in fact if with we have, from (42), that

The boundness of follows directly from (42). Now suppose that are such that , with then again from (42) we get

where

Then is absorbing for the semigroup .

Proposition 6. For all one has that the set

is absorbing for the semigroup .

Proof. We have from (42) and (43) that

Now, take such that , with . Then, by (42) and (44) we get that

Therefore,

Consequently, we have that

where

Now using the equivalence of the norm on and (52) we get

Let . If is such that

then we have that

where

Thus, we have proved the following.

Proposition 7 7. For all one has that the ball

is an absorbing set for the semigroup in .

4.2. Absorbing Set in

Now suppose that the initial data has some additional regularity. Then we can prove the existence of more regular absorbing sets. Let us define the following space:

equipped with the norm

Then we have the following.

Proposition 8 8. There exists such that the closed ball

is a bounded absorbing set for in .

Proof. If we multiply (1) by and integrate over we obtain

Let us denote the differential term of the previous inequality as

We will estimate the right hand side of (63). The first term can be estimated as follows:

Let us define

then we can rewrite the the second term of r.h.s of (63)

and estimate

Then if and setting , we have

To conclude the proof we note that

and that

From the previous inequalities we get

Then from (72) we get

and by Gronwall's lemma we obtain

Then if such that

we have that

where

and this concludes the proof.

4.3. Absorbing Set in

We consider the space as defined in (7), equipped with the norm defined in (10). By considering more regular initial conditions we can prove the following result.

Proposition 9 9. There exists such that the closed ball

is a bounded absorbing set for the semigroup in .

Proof. Multiply (1) by ; then we obtain

We call the differential term of the previous inequality and we estimate the r.h.s.:

Let us define

then

We estimate the last two terms of the previous equality:

Then if we set we get

Moreover we have

Then putting all together we get

From the previous inequality we get

and by Gronwall's lemma

Using again (86) we get

Then if such that

then there exists :

where

and this concludes the proof.

5. Global Attractor

In this section we will show the existence of a global attractor for the semigroup in . Since we have already proved the existence of the absorbing set in , then it is sufficient (see, e.g., [4] or [14] for general results or [15] for a recent application on a weakly damped wave equation) to prove that, for any fixed bounded set , the solution semigroup admits the decomposition:

such that

Let be a fixed bounded set and let . We will define the decomposition of as follows:

where and are solutions of the following problems:

where

Before showing that the semigroups and satisfy the conditions (94) and (95), respectively, we consider the following lemma (see [9]) that will be useful for the sequel.

Lemma 10 10. Let be an absolutely continuous function which fulfills, for some and almost every , the differential inequality

where is a positive function satisfying

Then

5.1. The Semigroup

Due to the similarity of problem (97) to (1), Proposition 9 still holds in this setting. We will omit the proof. Consequently we get that condition (94) holds by noting that .

Moreover by (97), multiplying the equation by and integrating over , we get that there exists a constant such that

Then integrating the above equation on and using (30) we get that there exist constants and such that

Consequently, for any there exists a constant such that

5.2. The Semigroup

Let be a parameter to be determined later and let us multiply, in , (98) by . Then we get

We estimate the term involving and its derivatives:

where is between and and satisfies . Moreover we have

where is between and , is between and and . Moreover we have used that .

We denote as the differential term of (106) and prove that it induces a norm that is equivalent to that of . Then from (99) and (107) we have that

Moreover we have

where we have used the same argument as (107) and (108) and where

Then we have

We estimate the right hand side of (106):

where

Then it remains to estimate the following term:

Then, using the above estimates, we can rewrite (106) in the following way:

Then by Lemma 10 and from (105) we conclude that there exists a positive constant such that

and consequently from (112) we conclude

This shows that the semigroup satisfies condition (95) and hence we have proven the following.

Theorem 11 11. The semigroup possesses a connected global attractor that is bounded in .

Acknowledgment

The authors would like to thank the editor and the referees for their helpful suggestions and remarks. This work was partially supported by projects PPTA 4476 and PPTA 5571 of Pontificia Universidad Javeriana, Bogotá.

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