# 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

##### Abstract

In this article, we start by performing a simple conversion of equation (1) to equation (2), then Liouville-type theorem is described afterward. We take in consideration the well-known high order semi linear parabolic equations with a non-linear infinite memory term. In third part of this article, we review the elements in which we prove the local existence of such equations with an infinite memory of nonlinear terms.

##### Keywords

**. 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).