On a fractional Cauchy problem with singular initial data

Abstract

Abstract This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mi>x</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {x}_{0} is the singular generalized function and F satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> </m:math> {L}^{\infty } logarithmic type, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {D}^{\alpha } is the Caputo derivative of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>α</m:mi> <m:mo>&lt;</m:mo> <m:mi>m</m:mi> </m:math> m-1\lt \alpha \lt m with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:math> m\in {{\mathbb{N}}}^{* } , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.