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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 749683, 18 pages
Semilinear Parabolic Equations on the Heisenberg Group with a Singular Potential
1Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, UMR 6085 CNRS, Avenue de l'Université, BP 12, 76801 Saint Etienne du Rouvray, France
2Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre Cedex, France
Received 29 October 2011; Accepted 12 December 2011
Academic Editor: Shaher M. Momani
Copyright © 2012 Houda Mokrani and Fatimetou Mint Aghrabatt. 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.
We discuss the asymptotic behavior of solutions for semilinear parabolic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We also establish the existence of a global branch of the corresponding steady states via the classical Rabinowitz theorem.
In this paper, we study a class of parabolic equations on the Heisenberg group . Let us recall that the Heisenberg group is the space of the (noncommutative) law of product
The left invariant vector fields are
In the sequel we will denote, we will denote and for . We fix here some notations:
where is the Heisenberg distance. Moreover, the Laplacian-Kohn operator on and Heisenberg gradient is given by
Let be an open and bounded domain of , we define thus the associated Sobolev space as follows
and is the closure of in .
We are concerned in the following semilinear parabolic problem
D’Ambrosio in  has proved Hardy’s inequality: let , it holds that
And by the work of Dou et al. , we have the following Hardy inequality with remainder terms for all :
for any , where , and . Moreover is optimal and it is not attained in .
Stimulated by the recent paper in the Euclidean space of Karachalios and Zographopoulos  which studied the global bifurcation of nontrivial equilibrium solutions on the bounded domain case for a reaction term , where is a bifurcation parameter; our focus here is devoted to some results concerning the existence of a global attractor for the (1.6) and the existence of a global branch of the corresponding steady states
with respect to . Let us recall some definitions on semiflows.
Definition 1.1. Let E be a complete metric space, a semiflow is a family of continuous maps , , satisfying the semigroup identities
For and ,
The positive orbit of through is the set
then the positive orbit of is the set . The -limit set of is
The -limit set of is
The subset attracts a set if . is invariant if and for all .
The functional is a Lyapunov functional for the semiflow if(i) is continuous.(ii) for .(iii) is constant for some orbit and for all .We have the following theorem from Ball [6, 7].
Theorem 1.2. Let be an asymptotically compact semiflow, and suppose that there exists a Lyapunov functional . Suppose further that the set is bounded, then is dissipative, so there exists a global attractor .
For each complete orbit containing lying in , the limit sets and are connected subsets of on which is constant.
If is totally disconnected (in particular countable), the limits exist and are equilibrium points. furthermore, any solution tends to an equilibrium point as .
The outline of the paper is as follows. In Section 2, we study the existence of an unbounded connected branch of positive solutions of (1.10) with respect to the parameter by using global bifurcation theorem introduced by López-Gómez and Molina-Meyer in . In Section 3, we describe the asymptotic behavior of solutions of (1.6) when has low energy smaller than the mountain pass level.
2. Existence of a Global Branch of the Corresponding Steady States
From the study of spectral decomposition of with respect to the operator , where the singular potential satisfies Hardy’s inequality (1.8), we have the following.
Proposition 2.1. Let . Then there exist , such that for each , the following Dirichlet problem admits a nontrivial solution in . Moreover, constitutes an orthonormal basis of Hilbert space .
For the proof of this proposition, we refer to .
Remark that the first eigenvalue characterized by is simple with a positive associated eigenfunction .
We discuss the behavior of when and .
Proposition 2.2. Let and . Then, (i) is a decreasing sequence, and there exist such that .(ii)The corresponding normalized eigenfunction converging weakly to 0, in .
Proof. (i) Let . The characterization (2.2) of implies that .
The improved Hardy inequality (1.9) implies that is bounded from below by . So, there exist such that .
(ii) The eigenfunction satisfies, for any : We still denote by the sequence of normalized eigenfunction, forming a bounded sequence in . Then there exists such that For some fixed small enough and any , we have Thus, We assume that , so passing to the limit in (2.3), we get that is a nontrivial solution of the problem However, is not achieved in , so .
Thanks to Hardy’s inequality (1.8) and Poincaré’s inequality, is equivalent to the norm on for all , so that we will use as the norm of .
Theorem 2.3. Let a bounded domain and assume that . Then, there exists an unbounded component of the set of positive solutions of (1.10) bifurcating from .
Proof. We introduce the Banach space , and the inner product in is given by
The bilinear form is continuous in , so the Riesz representation theorem implies that there exist a bounded linear operator such that
The operator is self-adjoint and compact and its largest eigenvalue is characterized by
We define the following energy functional on :
Similar to the classical case, is well defined on and belongs to , and we have for any . Let , is the dual space of , defined as by for all . Since is a bounded linear functional, is well defined, and , where ,
So, By Sobolev embedding Sobolev theorem , we have Then,
Consequently, hypotheses (HL) and (HR) of  are hold. If is a nonnegative solution of (1.10), then it follows from the strong maximum principle of J.-M. Bony  and the generalization of the Hopf boundary point lemma on the Heisenberg group , that lies in the interior of the cone: Hence, the assumption (HP) of  is fulfilled.
Remark 2.4. According to the theory of Rabinowitz , we can see that there is a continuum of the set of nontrivial solutions of (1.10), and the continuum consists of two subcontinua and . However, this does not necessarily implies that the subcontinuum satisfies the global alternative of Rabinowitz  by the reasons already explained by Dancer , López-Gómez and Molina-Meyer [8, 15]. Instead, the existence of a global subcontinuum of the set of positive solutions follows by slightly adapting [8, Theorem 1.1].
3. Asymptotic Behavior of Solutions for Problem (1.6)
Similar to [16, 17], we are interested here in the description of the behavior of solutions of (1.6) when has low energy smaller than the mountain pass level In view of , since , the functional satisfies the Palais-Smale condition and admits at least a positive solution (called mountain pass solution).
Proposition 3.1. Let , , and , the problem (1.6) has a unique weak solution such that and we have
Proof. By means of the Hill-Yosida theorem, is the semigroup generated by the operator . Let the function defined by , for . Since is locally Lipschitz, so by Pazy [18, Theorem 1.4] or Cazenave and Haraux [19, Theorem 6.2.2], there exists a unique solution of (1.6) defined on a maximal interval , where and
satisfying the variation of constants formula
Moreover, if , we say that is blow-up time, whereas if , we say that is global solution.
We will show that satisfies (3.3): Let , ( is the domain of definition of ), and . Since , we have
Set , and let , such that Define , then, and satisfies Thus, from (3.6), Passing to the limit, we deduce (3.3).
Next, we introduce the following sets:
is named the Nehari manifold relative to . The mountain-pass level defined in (3.1) may also be characterized as
Theorem 3.2. If there exist such that , then blows up in finite time.
Proof. Let such that , and we suppose that is a global solution for the problem (1.6). Since satisfy (3.3), we have
Set , then
Hence, we get for , .
Let , so we deduce by (3.13), that for any :
Hence, for any sufficiently large, we have Then and so , which is a contradiction.
Theorem 3.3. Assume that and , then the problem (1.6) admits a global solution . Moreover, there exists a positive number such that
Proof. Let , and let be the unique solution established in Proposition 3.1. From inequality (3.3), we have that is strictly decreasing, so
Suppose there exists such that . Then,
Moreover, since the application is continuous, there exists such that
Hence, in or . If in , then by the uniquess of , we conclude that for any . Thus, is global by extending to 0 for all , and so for any by Theorem 3.2. But , which is a contradiction. So, we conclude that for all .
On other hand, we can write Since satisfy (3.3), we have Then we have which implies that is a global solution of the problem (1.6), and is invariant set. Letting in (3.23), the integral is finitely determined. Therefore, there exists a sequence with as such that Letting , we obtain that is a solution of problem (1.10). So If , then , and so Since satisfies (3.3), it follows by Hölder inequality and from (3.24), that Therefore, We deduce by (3.22), (3.25), and (3.28) that which contradicts (3.26), and so in . Hence, by (3.24), we have Since we have For simplicity, let us denote by the divergent sequence and by . We have from (3.29) that So, due to (3.3) we have Therefore, there exists such that for all , On the other hand, Let , we have Let us recall that if we set , then So we get from (3.22) that for any , we have So from (3.37) and (3.39), we have for any that Since , there exists such that for any , we have Thus, with . But we remark that hence, we deduce that for any , we have and we can conclude that for any , we have
Remark 3.4. for small , Theorem 3.3 is an immediate consequence from the fact that, according to the linearized stability principle, the trivial solution is linearly asymptotically stable. In other words, from the fact that the principle eigenvalue of the linearization at is positive.
Questions of stability for nonlinear systems are frequently resolved via linearized stability or Lyapunov-type methods. Here, we proved the asymptotic stability under Lyapunov function to obtain estimates in .
Corollary 3.5. Assume that and . Then any solution of (1.6) tends to the trivial equilibrium point, as .
Proof. It follows from (3.45) that the semiflow is eventually bounded, see . Since the resolvent of the operator is compact, is compact for (see [20, Theorem 3.3], thus by [18, Corollary 3.2.2], is asymptotically smooth and so by [7, Proposition 2.3] is asymptotically compact. It remains to shows that , the set of equilibrium points of , is bounded: , so . Then from (3.3) and Poincaré’s inequality, we have which implies that the set is bounded. Then, by Theorem 1.2, is dissipative and by (3.45), we have as , for every bounded set . So, we conclude that the global attractor , and that any solution tends to the trivial equilibrium point as , when .
Theorem 3.6. Assume that . Then the solution of the problem (1.6) blows up in finite time.
Proof. Let , and let be the unique solution, the existence of which has been proved in Proposition 3.1. From the inequality (3.3), we have that is strictly decreasing, so Suppose there exists such that . Then And since the application is continuous, there exists such that Hence, in or . If in , then by the uniquess of , we conclude that for any . Thus, is global by extending to 0 for all , and thanks to Theorem 3.2, for any . But , which is a contradiction, and so . But by , then , which contradicts (3.47). So, we conclude that for all . We suppose by contradiction that , that is, exists for all . For , we have Then is strictly increasing and so We suppose that . Following the same reasoning as in the proof of Theorem 3.3, we deduce that we can select a divergent subsequence, still denoted by , such that when , Letting in the inequality we get that , which is a contradiction. So we conclude that Set , so By Hölder inequality, we have and by (3.55), there exist and a constant such that for , we have Then, there exist and a constant such that for , we have Hence, we have from (3.59), that for any , which is a contradiction if is sufficiently large. So we conclude that .
The author is glad to thank the referee for a careful and very constructive reading of the paper and making many good suggestions.
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