Nonlinear Memory Term for Fractional Diffusion Equation

This paper examines the Cauchy problem described by the following equation: partial derivative t lambda+1 phi-Delta phi=integral 0t(t-s)-gamma phi(s,.)pds,phi(0,x)=phi 0(x),phi t(0,x)=phi 1(x).(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{t}<^>{\lambda +1}\phi -\Delta \phi =\int _{0}<^>{t}(t-s)<^>{- \gamma } \left| \phi (s,.) \right| <^>{p}ds,\quad \phi (0,x)=\phi _{0}(x),\quad \phi _{t}(0,x)=\phi _{1}(x). \quad \mathrm{(1)} \end{aligned}$$\end{document}The equation involves the Caputo fractional derivative in time, denoted as partial derivative t lambda+1 phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{t}<^>{\lambda +1}\phi$$\end{document}. Additionally, The nonlinear term is determined by the memory term integral 0t(t-s)-gamma phi(s,.)pds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{0}<^>{t}(t-s)<^>{- \gamma } \left| \phi (s,.) \right| <^>{p}ds$$\end{document}, where gamma is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document}. Using the fixed point theorem, we establish the global existence of solutions to the Cauchy problem (1) for small initial data. We also investigate the impact of the nonlinearity parameter on the range of the exponent p and the estimation of the solutions.