A subordination principle for subdiffusion equations with memory

Abstract

We study the existence of mild solutions to subdiffusion equations with memory ( ∗ ) ∂ t α u ( t ) = A u ( t ) + ∫ 0 t κ ( t − s ) A u ( s ) d s , t ≥ 0 , with the initial condition u(0)=x, where 0<α<1, A is a closed linear operator defined on a Banach space X, the initial value x belongs to X and κ is a suitable kernel in Lloc1(ℝ+). First, we find a subordination formula for the solution operator of (∗) and then we study its connection with the existence of mild solution to the first order diffusion equation with memory.