Some Study of a Cauchy Problem in Parabolic Integrodifferential Equations Class

Introduction

Let’s have the parabolic integrodifferential equations of the form

where  the  partial differential operator   

is uniformly elliptic,

is a family of linear bounded operators defined on the space of all square  integrable  functions L₂(E_n) and E_n is the n- dimensional Euclidean space.

We consider integrodifferential equations of the form;

Where,

is an n-dimensional multi index,

is an element of the n-dimensional Euclidean space E_n.  Let L₂(E_n) be the space of all square integrable function on E_n and W^m(E_n)  the Sobolev space, [ the space of all functions 

such that he distributional derivative

 with 

all belong to 

We assume that u_t  satisfy the Cauchy condition;

We can assume that on g(x)=0 on E_n. We shall say that u_t is of the class S if for each

where

 is the abstract derivative of u_t in L₂(E_n) in other word there is and element 

such that

Where

is the norm in 

Notice that if 

for each 

  then the partial derivative 

exists in the usual sense

 and 

in fact according to the embedding theorem [3], [8], we have

Where

is a positive constant,

is the volume enclosed by the sphere 

 letting 

we get 

on Q x (0, T)

Since b is arbitrary, it follows that

Let us suppose that the following assumptions are satisfied; 

The coefficients 

 are real functions of t, defined on [o,T] and having continuous derivatives

 on [o, T]

The deferential operator 

is uniformly elliptic on [o, T]

The kernels

 

are linear bounded operators acting on into it self. It is assumed that these operators are continuous in

Furthermore it is assumed that the (abstract) partial derivatives 

 exist  for all 

and represents linear bounded operators on L₂(E_n) which are continuous in

The coefficients 

are real functions, which are continuous  and bounded on 

is a map from [0,T] into which is continuous in t with respect to the norm in

All the coefficients 

have continuous bounded partial derivatives

The range of the operators 

is the space

 we assume that all the operators 

are bounded, and that 

 exist for all 

The operators

 are supposed to be bounded and continuous in 

 for all

 It is supposed also that 

are continuous in 

Proposition 1.

Under conditions (a) , …. , (e) if there is at least one solution in the class S of the Cauchy problem (*), (**), then this solution is the unique such solution.

Proof.

 If 

then we have the following representation:

Where

is the singular integral operator defined to be the

and

Notice that 

 are bounded operators from L₂(E_n) into itself, [1],[2].

Let H₁(t)  be an operator defined by

According to assumption, the operator H₁(t) has a bounded inverse 

defined on for every

Set

and

Using (1), then equation (*) can be written in the form;

To Prove the uniqueness of the considered Cauchy problem, we set

Now set

Then according to assumptions (1) and (2), we can write

Where G is the fundamental solution of the Cauchy problem for the parabolic equation

Let

be the family of bounded operators defined by

Consequently (3) can be written in the form

According to the well-known properties of the fundamental solution G, [4], [5], we can see that

For

 

where C is a positive constant and is a constant satisfying 0<y<1. Substituting from (4) into (2), we get

Using (5) and (6), we get from (7), the following estimation;

(To obtain (7) and (8), we already used conditions (c) and (d) where C is a positive constant. Thus (4) and (8) lead immediately to the fact that u_t(x)=0 on E x [0,T].

Proposition2

Under the condition (a) , …, (h) the solution of the Cauchy problem (*) , (**) exists in the class S.

Proof
Using the conditions from (a) to (e), we obtain

According to (5) and (6), the Volterra integral equation (9) has a unique solution t V in which satisfies:

where c is a positive constant.

This means that under conditions from (a) to (e), we can obtain the so called mild solution [6] of the Cauchy problem (*) , (**). this solution is represented by

Now we must prove that the distributional derivatives

 exists in L₂(E_n) for all 

To prove that 

 

we apply formally the differential operator on both sides of the integral equation (9), then we get;

Let us consider as the unknown element in the integral equation (10). Under the assumptions (a),…, (h) this integral equation can be solved for

Thus

for

Now we have

Using (11), we can wire

Where

Proposition 3

Let 

L₂(E_n) for every 

and continuous in t. suppose.

Where

 is continuous in 

if is a sequence of functions of the class S, which are solutions of the Cauchy problem

then

 converges in L₂(E_n) to the solution of the Cauchy problem (*), (**).

Proof. Set

We find

References

1. A.S. Al-Fhaid, The Derivation of Interstellar Extinction Curves and Modelling, Ph.D. Thesis.
2. A. S. Al-Fhaid, Continued fraction Evaluation of the Error function. To appear.
3. Calderon H. P., Zygmond A. Singular Integral Operators and Differential Equations II, Amer J. Math., 79,(3); (1957).
4. EhrenperiesL, Cauchy’s Problem for linear Differential Equations with constant coef-ficients II, Proc. Nat. Acad. Soi USA, 42. Nv. 9.
5. El-Borai. M. On the correct formulation of the Canchy problem Vesnik Moscow univ. 4 (1968).
6. Friedman A. Partial differential Equations, Robert E. Krieger Publishing Company, New York, 101-102 (1976).
7. Hormander L. , Linear partial differential Operators ( Springer – Verlag) , Berlin, New York (1963).
   CrossRef
8. Samuel M. R., Semilinear Evolution equations in Banach spaces with applications of Parabolic partial differential Equations, Trous Amer. Math. Soc. 33: 523-535 (1993).
9. Yosida K. Functional Analysis, Spreinger verlage, (1974).