Nonlinear Memory Term for Fractional Diffusion Equation

This paper examines the Cauchy problem described by the following equation: \(\begin{aligned} \partial _{t}^{\lambda +1}\phi -\Delta \phi =\int _{0}^{t}(t-s)^{- \gamma } \left| \phi (s,.) \right| ^{p}ds,\quad \phi (0,x)=\phi _{0}(x),\quad \phi _{t}(0,x)=\phi _{1}(x). \quad \mathrm{(1)} \end{aligned}\) $$\begin{array}{c}\hfill {\partial }_t^{\lambda +1}\varphi -\mathrm{\Delta }\varphi ={\int }_0^t{(t-s)}^{-\gamma }{\left|\varphi ,(,s,,,.,)\right|}^{p}ds,\varphi (0,x)={\varphi }_0(x),{\varphi }_t(0,x)={\varphi }_1(x).(1)\end{array}$$ The equation involves the Caputo fractional derivative in time, denoted as \(\partial _{t}^{\lambda +1}\phi\) ${\partial }_t^{\lambda +1}\varphi$ . Additionally, The nonlinear term is determined by the memory term \(\int _{0}^{t}(t-s)^{- \gamma } \left| \phi (s,.) \right| ^{p}ds\) ${\int }_0^t{(t-s)}^{-\gamma }{\left|\varphi ,(,s,,,.,)\right|}^{p}ds$ , where \(\gamma \in (0,1)\) $\gamma \in (0,1)$ . 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.