A. S. Al-Fhaid
Department of Mathematics, College of Sciences, P.O.Box 80203 King Abdulaziz University, Jeddah - 21589, Saudi Arabia.
DOI : http://dx.doi.org/10.13005/msri/070111
Article Publishing History
Article Received on : 26 Mar 2010
Article Accepted on : 28 Apr 2010
Article Published :
Plagiarism Check: No
Article Metrics
ABSTRACT:
We consider the parabolic integrodifferential equations of a form given below. We establish local existence and uniqueness and prove the convergence in L2(En) to the solution Ut of the Cauchy problem.
KEYWORDS:
Cauchy; Parabolic; Integrodiffrential equationss; Operator
Copy the following to cite this article:
AL-FHAID A. S. Some Study of a Cauchy Problem in Parabolic Integrodifferential Equations Class. Mat.Sci.Res.India;7(1)
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Introduction
Let’s have the parabolic integrodifferential equations of the form
![eq a](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqa.jpg)
![eq b](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqb.jpg)
where the partial differential operator
![eq c](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqc.jpg)
is uniformly elliptic,
![eq d](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqd.jpg)
is a family of linear bounded operators defined on the space of all square integrable functions L2(En) and En is the n- dimensional Euclidean space.
We consider integrodifferential equations of the form;
![eq g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqg.jpg)
Where,
![eq i](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqi.jpg)
is an n-dimensional multi index,
![eq j](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqj.jpg)
![eq h](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqh.jpg)
![eq k](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqk.jpg)
is an element of the n-dimensional Euclidean space En. Let L2(En) be the space of all square integrable function on En and Wm(En) the Sobolev space, [ the space of all functions
![eq o](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqo.jpg)
such that he distributional derivative
![eq p](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqp.jpg)
with
![eq q](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqq.jpg)
all belong to
![eq r](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqr.jpg)
We assume that ut satisfy the Cauchy condition;
![eq s....](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqs....jpg)
We can assume that on g(x)=0 on En. We shall say that ut is of the class S if for each
![eq v](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqv.jpg)
where
![eq x.](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqx..jpg)
is the abstract derivative of ut in L2(En) in other word there is and element
![eq z](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqz.jpg)
such that
![eq a1](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eqa1.jpg)
Where
![eq 2b](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2b.jpg)
is the norm in
![Vol7_No1_Som_A.-S.-A_eq2b'](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2b1.jpg)
Notice that if
![eq 2c](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2c.jpg)
for each
![eq 2d](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2d.jpg)
then the partial derivative
![eq 2d'](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2d1.jpg)
exists in the usual sense
![eq 2e](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2e.jpg)
and
![eq 2f](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2f1.jpg)
in fact according to the embedding theorem [3], [8], we have
![eq 2g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2g.jpg)
Where
![eq 2h](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2h.jpg)
is a positive constant,
![eq 2g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2i.jpg)
![eq 2j](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2j.jpg)
is the volume enclosed by the sphere
![eq 2j'](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2j1.jpg)
letting
![eq k](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2k.jpg)
we get
![eq 2l](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2l.jpg)
on Q x (0, T)
Since b is arbitrary, it follows that
![eq 2m](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2m.jpg)
Let us suppose that the following assumptions are satisfied;
The coefficients
![eq 2n](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2n1.jpg)
are real functions of t, defined on [o,T] and having continuous derivatives
![eq o](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2o.jpg)
on [o, T]
The deferential operator
![eq 2q](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2q.jpg)
is uniformly elliptic on [o, T]
The kernels
![eq 2r](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2r.jpg)
are linear bounded operators acting on into it self. It is assumed that these operators are continuous in
![eq 2s](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2s.jpg)
Furthermore it is assumed that the (abstract) partial derivatives
![eq 2t](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2t.jpg)
exist for all
![eq 2u](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2u1.jpg)
and represents linear bounded operators on L2(En) which are continuous in
![eq 2w](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2w1.jpg)
The coefficients
![eq 2x](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2x.jpg)
are real functions, which are continuous and bounded on
![eq 2y](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2y.jpg)
![Vol7_No1_Som_A.-S.-A_eq2y'](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2y1.jpg)
is a map from [0,T] into which is continuous in t with respect to the norm in
![eq 2yy](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2yy.jpg)
All the coefficients
![eq 2z](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2z.jpg)
have continuous bounded partial derivatives
![eq 3a](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3-a.jpg)
![eq 3b](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3b.jpg)
The range of the operators
![eq 3c](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3c.jpg)
is the space
![eq 3d](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3d.jpg)
we assume that all the operators
![eq 3e](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3e.jpg)
are bounded, and that
![eq 3f](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3f.jpg)
exist for all
![eq 3g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3g.jpg)
The operators
![eq 3g'](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3g1.jpg)
are supposed to be bounded and continuous in
![eq 3i](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3i.jpg)
for all
![eq 3k](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3k.jpg)
It is supposed also that
![eq 3l](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3l.jpg)
are continuous in
![eq 3m](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3m.jpg)
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
![eq 3n](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3n.jpg)
then we have the following representation:
![eq 1](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq1.jpg)
Where
![eq 3o](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3o.jpg)
is the singular integral operator defined to be the
![eq 3p](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3p1.jpg)
and
![eq 3s](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3s.jpg)
Notice that
![eq 3t](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3t.jpg)
are bounded operators from L2(En) into itself, [1],[2].
Let H1(t) be an operator defined by
![eq 3w](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3w.jpg)
According to assumption, the operator H1(t) has a bounded inverse
![eq 3y](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3y.jpg)
defined on for every
![eq 3z](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3z.jpg)
Set
![eq 4a](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4a.jpg)
and
![eq 4b](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4b.jpg)
Using (1), then equation (*) can be written in the form;
![eq 2](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq2.jpg)
To Prove the uniqueness of the considered Cauchy problem, we set
![eq 4d](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4d.jpg)
Now set
![eq 4e](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4e.jpg)
Then according to assumptions (1) and (2), we can write
![eq 3](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq3.jpg)
Where G is the fundamental solution of the Cauchy problem for the parabolic equation
![eq 4f](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4f.jpg)
Let
![eq 4g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4g.jpg)
be the family of bounded operators defined by
![eq 4h](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4h.jpg)
Consequently (3) can be written in the form
![eq 4](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4.jpg)
According to the well-known properties of the fundamental solution G, [4], [5], we can see that
![eq 5](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5.jpg)
![eq 6](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq6.jpg)
For
![eq 4i](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4i.jpg)
where C is a positive constant and is a constant satisfying 0<y<1. Substituting from (4) into (2), we get
![eq 7](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq7.jpg)
Using (5) and (6), we get from (7), the following estimation;
![eq 8](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq8.jpg)
(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 ut(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
![eq 9](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq9.jpg)
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
![eq 4o](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4o.jpg)
Now we must prove that the distributional derivatives
![eq 4p](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4p.jpg)
exists in L2(En) for all
![eq 4r](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4r.jpg)
To prove that
![eq 4s](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4s.jpg)
we apply formally the differential operator on both sides of the integral equation (9), then we get;
![eq 10](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq101.jpg)
Let us consider as the unknown element in the integral equation (10). Under the assumptions (a),…, (h) this integral equation can be solved for
![eq 4z'](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4z.jpg)
Thus
![eq 4z](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq4z1.jpg)
for
![eq 5a](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5a.jpg)
Now we have
![eq 11](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq11.jpg)
Using (11), we can wire
![eq 5b](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5b.jpg)
Where
![eq 5c](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5c.jpg)
Proposition 3
Let
![eq 5d](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5d.jpg)
L2(En) for every
![eq 5g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5g.jpg)
and continuous in t. suppose.
![eq 5f](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5f.jpg)
Where
![eq fm](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5m.jpg)
is continuous in
![eq g](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5g1.jpg)
if is a sequence of functions of the class S, which are solutions of the Cauchy problem
![eq 5h](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5h.jpg)
then
![eq 5i](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5i.jpg)
converges in L2(En) to the solution of the Cauchy problem (*), (**).
Proof. Set
![eq 5k](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5k.jpg)
We find
![eq 5l](http://www.materialsciencejournal.org/wp-content/uploads/2010/02/Vol7_No1_Som_A.-S.-A_eq5l.jpg)
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CrossRef
- Samuel M. R., Semilinear Evolution equations in Banach spaces with applications of Parabolic partial differential Equations, Trous Amer. Math. Soc. 33: 523-535 (1993).
- Yosida K. Functional Analysis, Spreinger verlage, (1974).
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