Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations non-homogenous and values. consider both weak strong solutions problems. For solutions, we introduce a definition of which allows to prove existence solution non-zero values non-homogeneous source terms lying in some negative-order Sobolev spaces. an optimal compatibility condition solutions. also sharp conditions guaranteeing more regularity space.