The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.


Introduction
Let us consider the following singular evolution equations: (1.2) Where { (t)}: t  J, i = 1,2,…, r} is a family of linear closed operators defined on densely domains in a Banach space E, J = [0 , T] is an interval, T > 0, {K i (t, ), t,   J,  < t} is a family of linear bounded operated defined on E to E, such that For all h  E, 0   1, M is a positive constant independent on t and , . is the norm in E, V (t, ) = (K 1 (t, ) A 1 ()u(), …, K r (t, ) A r () u()), F is an abstract function defined on E r to E, for every t,   J, t > , f(t)  E is a given continuous function in t  J.
Let D (A i ), i = 1,…, r be the domain of definitions of A i . It is supposed that the domains D(A 1 ),…, D(A r ) are independent of t.
We assume that all the functions A 1 (t) h, … , A r (t) h are continuous on J for every h  ∩ =1 D ( ) and all the functions K 1 (t, ) h, …, Kr (t, ) h are continuous on J  J, t > , for every h  E.
In section 2, we shall define the abstract parabolic transform. Using this transform, we can find a dense set S in E such that if f(t)  S, then equation (1.1) can be solved.
In section 3, we shall solve fractional integral evolution equations and Hilfer fractional integral evolution equations.
Also, some general singular integro-partial differential equations are studied. The properties of the solutions in all cases are given. In section 4, we shall study equation (1.2).

Abstract Parabolic Transforms
The parabolic transform is defined in [14]. Let us, now, try to modify the definition to be suitable for abstract integral equations. Let Q(t) be a strongly continuous semi-group defined on E, with infinitesimal generator B defined on dense set D (B) in E, such that D(B)  ∩ =1 D ( ).
Let us study the following of singular integral evolution equation; Let { n } be a sequence defined by: The zero approximation  0 (t) is chosen to be identically zero. Using conditions (1.3) and (2.1) and the properties of the operators A, K, Q, it can be proved that the functions  1 (t), …,  n (t), … are continuous in t  J and with values in E.
Again, according to (1.3) and (2.1), we get Where (.) is the gamma function. The last inequality leads to the fact that the sequence { n (t)} uniformly converges in E to a continuous function (t) on J, which represents the solution of equation (2.3). To prove the uniqueness, let us suppose that there are two solutions (t) and  * (t) of (2.3). Thus; The following theorem proves that the solution (t) of equation (2.3) depends continuously on f(t) The last inequality leads to the required result.
Let us now try to discuss the ill-posed problem (1.1). For this purpose, we need some of additional conditions. Suppose that for every i = 1, …., r and every 1 , 2 , 3  J, the operator Q(t 1 ) commutes with K i (t 2 , t 3 ) and commutes with A i (t 3 ) : and using conditions (2.4) and (2.5), we find that for every n, u n (t) satisfies the equation Notice that according to theorem (2.2), u n (t) depends continuously on f(t).
As a direct application, we consider the following abstract Hilfer fractional integral equation: Equation (2.6) can be solved if we replace g by Q ( 1 ) g.
The coefficients 1 , …. are real continuous functions on J, for all | |  m.
For t > , the kernels K 1 (t, ), ..., K r (t, ) are real continuous functions such that; M is a positive constant, 0 <   1, t > , t,   J. Let C m (R k ) be the set of all real continuous functions on R k , which have continuous derivatives of order less than or equal to m.
Let W m (R k ) be the completion of C m (R k ) with respect to the norm: ‖ ‖ 2 = ∑ | | ≤ m ∫ | g ( )| 2 dx.
Let E = L 2 (R k ) be the space of all square integral functions on R k .
It is assumed that f is a real function defined on R k such that for every t  J, f  L 2 (R k The strongly continuous semi group Q(t) with the infinitesimal generator B is given by Where; G is the fundamental solution of the following equation For sufficiently large N, we can find constants M > 0, Where . is the norm in L 2 (R k ), The domain of definitions of the operators A i (t) = ∑ | |≤ ( )D and B are W m (R k ) and W N+2 (R k ) respectively.
These domains of definitions are dense in L 2 (R k ) and the operators A i (t) and B are closed in L 2 (R k ).
It is clear that the operators A 1 () , .., A r () are commute with Q (t), for all t,   J.
Notice also that equations (3.1) and (3.4) are solved without any restrictions on the characteristic forms of the principle parts of the partial operators Let us study the following equation: where; K is continuous for t >  and satisfies condition (3.2) and f satisfies the conditions as in theorem (3.1). let B be the operator defined by: , Let us suppose some regularity conditions on a k ; D q a q  C b (R k ), q  N * , for sufficiently large N * , N * > 2m (2N+1), where; C b (R k ) is the set of all continuous bounded functions on R k .
Let us suppose also that For all x  R k , where M is a positive constant, | | 2 = y 1 2 + .... + 2 The operators A and B are closed operators in L 2 (R k ) with domain of definitions W 2m (R k ) and W 2m(2N+1) (R k ) respectively.
The operator B = A 2N+1 is the infinitesimal semi-group Q(t) defined by Where G is the fundamental solution of the equation According to the properties of the fundamental solution G, we deduce that the operator Q satisfies condition (3.3), (see [19][20][21][22] Thus; Notice that equation (3.5) is still ill-posed, but we can now solve the following equation: , for every t  J and every n = 1, 2, … ( n will be determined as in theorem (2.2)).

Nonlinear Equations
The results in sections 2 and 3 are obtained after suitable modification on the given function f, which plays in some special cases the role of initial condition.
Let us consider in the Banach space E, the following equation; Theorem 4.1 under the conditions on f, F, Q, K i , A i of sections 1 and 2, (but without conditions (2.4) and (2.5)), there exists a unique solution (t)  E of equation (4.1). This solution is continuous in t  J and is continuously depending on f, (with respect to the norm in E).
Proof: Using condition (1.4), we find that the proof is similar to the proof in theorems (2.1) and (2.2).

Conclusions
A general singular integral evaluation equation is studied in Banach space. The abstract parabolic transform is constructed to investigate a wide class of ill-posed problems. The Hilfer fractional integro-partial differential equations are studied without any restrictions on the characteristic forms.