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Hyperbolic Relaxation of a Fourth Order Evolution Equation
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 .
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  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 ) 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 , the third time scale was studied; the authors proved the existence of a global attractor (see ) 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 . Moreover the authors proved the existence of an exponential attractor with finite fractal dimension of order . In  the authors proved the existence of an inertial manifold (see ) whose dimension is of order , and by the -dimensional volume elements methods (see ) an estimate of the dimension of the global attractor of order was found. This estimate is also consistent with the numerical experiments developed in ; 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  while in  the existence of a family of exponential attractors was proved. The viscous and nonviscous perturbation has been studied in  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 . 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.
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  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.,  or  for general results or  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 ) 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: