A semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions

We investigate a semilinear problem for a fractional diffusion equation with variable order Caputo fractional derivative \(\left( \partial _t^{\beta (t)} u\right) (t)\) $\left({\partial }_t^{\beta (t)},u\right)(t)$ subject to homogeneous Dirichlet boundary conditions. The right-hand side of the governing PDE is nonlinear (Lipschitz continuous) and it contains a weakly singular Volterra operator. The whole process takes place in a bounded Lipschitz domain in \({{\mathbb {R}}}^d\) ${\mathbb{R}}^d$ . We establish the existence of a unique solution in \(C\left( [0,T],L^{2} (\varOmega )\right) \) $C\left([0,T],,,L^2,(\Omega )\right)$ if \(u_0\in L^{2} (\varOmega )\) $u_0\in L^2(\Omega )$ . Moreover, if \(\mathcal {L}^{\gamma }u_0\in L^{2} (\varOmega )\) ${\mathcal{L}}^{\gamma }{u}_0\in L^2(\Omega )$ for some \(0<\gamma <1-\frac{\delta }{\beta (0)}\) $0<\gamma <1-\frac{\delta }{\beta (0)}$ ( \(\delta \) $\delta$ depends on the right-hand-side of the PDE) then \(\mathcal {L}^{\gamma }u\in C\left( {[}0,T{]},L^{2} (\varOmega )\right) \) ${\mathcal{L}}^{\gamma }u\in C\left([,0,,,T,],,,L^2,(\Omega )\right)$ .