A subordination principle for subdiffusion equations with memory

Abstract

We study the existence of mild solutions to subdiffusion equations with memory

$$\left(\ast \right){\partial }_t^{\alpha }u\left(t\right)=Au\left(t\right)+{\int }_0^t\kappa \left(t-s\right)Au\left(s\right)ds,t\ge 0,$$

with the initial condition $u\left(0\right)=x$, where $0<\alpha <1$, $A$ is a closed linear operator defined on a Banach space $X$, the initial value $x$ belongs to $X$ and $\kappa$ is a suitable kernel in $L_{loc}^1\left({ℝ}_{+}\right)$. First, we find a subordination formula for the solution operator of $\left(\ast \right)$ and then we study its connection with the existence of mild solution to the first order diffusion equation with memory.