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 (partial derivative(beta(t))(t) u) (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 unique solution in C([0, T], L-2(Omega)) if u(0) is an element of L-2(Omega). Moreover, if L-gamma u(0) is an element of L-2(Omega) takes place in a bounded Lipschitz domain in R d . We establish the existence of a for some 0 < gamma < 1 - delta/beta(0) (delta depends on the right-hand-side of the PDE) then L(gamma)u is an element of C ([0, T], L-2 (Omega)).