Existence of solutions to a fractional differential equation in the Heisenberg group

Existence of solutions to a fractional differential equation in the Heisenberg group

Abd elhakim Lamairia

Department of mathematic and in formatics, Tebessa-Algeria.

Corresponding Author Email: hakim24039@gmail.com

DOI : http://dx.doi.org/10.5281/zenodo.8378346

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. Introduction

In this paper, we investigate a differential equation of fractional order by reformulating it in the form of a higher-order semi-linear parabolic equation with non-local nonlinearity in time. Our equation is as follows:

1.1. Equation formula

                    (1)

where ,

(constant),
and the space  denotes the space of Hilbert, and is Heisenberg group. Equation (1) is equivalent to the equation:

We choose the constants , and  in the following way:  and , so we obtain the final form of the equation (1), which is

              (2)

Let us first present our well-posed Ness result.

1.2. Theorem

(Local existence). Given , There exist a maximal time  and a unique mild solution , to the problem (2).

2. Preliminaries

2.1. The Laplacian on the Heisenberg Group

If we identify  with the complex plane  via

and let

then  becomes a non-commutative group when equipped with the multiplication  given by

where  is the symplectic form of  and  defined by

and  is the complex conjugate of a complex number .

In fact, is a unimodular Lie group on which the Haar measure is just the ordinary Lebesgue measure .

Let  be the Lie algebra of left-invariant vector fields on . A basis for  is then given by  and , were

and

The Laplacian  on  is defined by

A simple computation gives

In this paper we need to give new estimates for the strongly continuous one parameter semigroup , generated by . More precisely, we use the Sobolev spaces , as in [1] and [2] to estimate , in terms of  for all  in , and to give an estimate for  in terms of .

The function  on  given by

is in fact the solution of the initial value problem

for the Laplacian .

Using the same techniques as in [1], we get for all  and ,

where , is given by

and  is the function on  given by

2.2. Lemma

Let , then for

is a bounded linear operator and

were

Proof: [See [8]. Theorem 5.5]

2.3. Definition

(Riemann-Liouville fractional derivatives)

Let  The RiemannLiouville left- and right-sided fractional derivatives of order  are, respectively, defined by

                 (3)

and

              (4)

 let  be the space of functions  which are absolutely continuous on

.

 and  In particular, ,

2.4. Definition

(Riemann-Liouville fractional integrals)

Let , The Riemann-Liouville left- and right-sided fractional integrals of order  are, respectively, defined by

               (5)

and

               (6)

Finally, taking into account the following integration by parts formula:

2.5. Proposition

For , we have the following identities

for all

and

for all , where .

3. Local Existence

This section is dedicated to proving the local existence and uniqueness of mild solutions to the problem (2). Let us start by the

3.1. Definition

(Mild solution). Let, and . We say that  is a mild solution of problem (2) if  satisfies the following integral equation

            (7)

where  and .

3.2. Proof of Theorem

For arbitrary ,

let , where we equip  with the following metric generated by the norm

Since  is a Banach space,  is a complete metric space. Next, for all , we define

We prove the local existence by the Banach fixed point theorem.

•  Let , Using [ lemma , we obtain for all , thanks to the following inequality

 is chosen such that

Then, by the Banach fixed point theorem, see e.g. there exists a mild solution , to problem (2).

for all .

Now, if we choose  small enough such that

               (8)

we conclude that , for all .

With

Therefore, using the fact that , and the continuity of the semigroup , we get .

4. –  is a Contraction:

For , using again [lemme  we have

4.0.1. Remark

for all ,thanks to the following inequality

                   (9)

Where  is a constant related to . here  is chosen such that

                    (10)

Then, by the Banach fixed point theorem, seee.g. There exists a mild solution  to problem (2).

• Uniqueness: if  are two mild solutions in  for some , using [ lemma  and (9), we obtain

for all . So, the uniqueness follows from Gronwall’s inequality (cf. [9]).

5. References

[1] Ahmed, N.U., Semigroup theory with application to systems and control. Logman Scientific, Tehnical, London, 1991.

[2]E. Lanconelli, F.Uguzzoni,Asymptotic behaviour and nonexistence theorems for semi -linear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group,Boll.Un. Math.Ital.,8,1998, p.139-168.

[3] E.Podlubny,Fractional Differential Equations, Asymptotic behaviour and nonexistence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group, Math. Sci. Engrg., 198, Academin Press, New York,1999.

[4] Local and global existence of solutions to semilinear parabolic initial value problems, S.B.Cui (cui Shangbin) Lanzhou University,Lanzhou,Gansu 730000,China

[5] N. Garofalo, E. Lanconelli, Existence and nonexistence results for semi-linear equations on the Heisenberg group, Indiana Univ. Math. J., 41, 1992, p.p. 71-97.

[6] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach Science Publishers, 1987

[7] S. I. Pohozaev, L. Véron, Apriori estimates and blow-up of solutions of semi-linear inequalities on the Heisenberg-group, Manuscripta Math., no. 1, p.p. 85-99.

[8] The Semigroup and the Inverse of the Laplacian on the Heisenberg Group1 APARAJITA DASGUPTA Department of Mathematics, Indian Institute of Science,Bangalore-560012, India email: adgupta@math.iisc.ernet.in

[9] T. Cazenave, A. Haraux, Introduction aux problèmes d’évolution semi-linéaires, Ellipses, Paris, (1990).