The Parabolic Transform and Some Singular Integral Evolution Equations

Mahmoud M El-Borai,Khairia El-Said El-Nadi

doi:10.13189/ms.2020.080406

Abstract

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.

Highlights

Let us consider the following singular evolution equations: u(t) = f(t) + ∫0t ∑ri=1 Ki (t, θ) Ai (θ) u (θ) dθ, (1.1)u(t) = f(t) + ∫0t F (V (t, θ)) dθ, (1.2) ‖Ki (t, θ)h ‖ ≤ M (t - θ)1-α ‖h‖, (1.3)For all h E, 0 1, M is a positive constant independent on t and, . is the norm in E, Let D (Ai), i = 1,..., r be the domain of definitions of Ai
It is supposed that the domains D(A1),..., D(Ar) are independent of t
We assume that all the functions A1(t) h, ... , Ar(t) h are continuous on J for every h ∩ri=1 D (Ai) and all the functions K1 (t, ) h, ..., Kr (t, ) h are continuous on J J, t >, for every h E

Summary

Introduction

For all h E, 0 1, M is a positive constant independent on t and , . Equations (1.1) and (1.2) are studied for a wide class of the closed operators A1(t),...,Ar(t). This means that we shall study ill-posed problems. Using this transform, we can find a dense set S in E such that if f(t) S, equation (1.1) can be solved.

Singular Integro-differential Equations

Nonlinear Equations

Conclusions