MIXED PROBLEM WITH A WEIGHTED INTEGRAL CONDITION FOR A PARABOLIC EQUATION WITH THE BESSEL OPERATOR

In this paper, we prove the existence, uniqueness and continuous dependence on the data of a solution of a mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator. The proof uses a functional analysis method based on an a priori estimate and on the density of the range of the operator generated by the considered problem. Singular Parabolic Equations, Weighted Integral Condition, A


Preliminaries
We first transform problem (1.1)-(1.4)with inhomogeneous boundary conditions (1.3) and (1.4) to an equivalent problem with homogeneous conditions by introducing a new unknown function defined as follows ?
Then problem (1.1)-(1.4)can be formulated as follows: Instead of searching for the function , we search for the function .So the solution of @ ? problem (1.1)-(1.4)will be given by .@Ð<ß >Ñ oe ?Ð<ß >Ñ  Y Ð<ß >Ñ For the investigation of the problem (2.We also use weighted spaces on the interval such as , and , whose definitions are analogous to those for functions defined on .For example, is the subspace of with the finite norm and is the Hilbert space of vector valued functions with We assume that satisfies the compatibility conditions of the form (2.3) and (2.4): : : :

Two-Sided A Priori Estimates
Theorem 1: For any function , we have a priori estimate ?− HÐPÑ From equation (2.1), it follows that Proof: The initial condition (2.2) implies that where is a positive constant independent of the solution .

Solvability of the Problem
It follows from inequality (3.1) that the operator is continuous.From inequality PÀ I Ä J (3.4), it follows that the range of is closed in .Therefore, there exists a continuous VÐPÑ P J inverse operator yielding the solution.That is, is a linear homeomorphism from the P P " space on the closed set .To prove that there exists a unique solution of problem Let the function be given by ?
U Now, we return to the proof of Theorem 3. It is sufficient to prove that the set is ´! .Hence .This completes the proof of Theorem 3.

Conclusion
In summary, we have established the existence, uniqueness and continuous dependence on given data for solutions to a mixed problem for a parabolic equation with the Bessel operator which combines inhomogeneous Neumann condition and integral conditions.More precisely, we have constructed a sufficiently smooth function satisfying these conditions.Then we are lead to study an equivalent problem with homogeneous boundary conditions.Thus, we have established two sided a priori estimates for the operator generated PÀ I Ä J by the considered problem.Then we concluded that the operator realizes a linear P homeomorphism of the space on the closed set .To prove that the studied I V Ð P Ñ § J problem possesses a unique solution, we proved that is dense in .VÐPÑ J

F
4), we have to show that .where is a positive constant independent of the solution .-? To establish the proof of this theorem, we need the following: Proof: Proposition: If, for all in the set and for some function ?H ÐPÑÀ oe Ö? − HÐPÑÀ j? oe !Using the fact that (4.1) holds for any function , we can Proof of Proposition: ?− H ÐPÑ !express it in a particular form.Let us define the function by the relation * * = Ð<ß >Ñ oe Ð<ß =Ñ.=ß ' X > and let be a solution of the equation ?