On well-posedness of the sub-diffusion equation with conformable derivative model

Abstract

In this paper, we study an initial value problem for the time diffusion equationC∂β∂tβu+Au=F,0<β≤1,on Ω × (0, T), where the time derivative is the conformable derivative. We study the existence and regularity of mild solutions in the following three cases with source term F:•F=F(x,t), i.e., linear source term;•F=F(u) is nonlinear, globally Lipchitz and uniformly bounded. The results in this case play important roles in numerical analysis.•F=F(u) is nonlinear, locally Lipchitz and uniformly bounded. The analysis in this case can be widely applied to many problems such as–Time Ginzburg-Landau equations C∂βu/∂tβ+(−Δ)u=|u|μ−1u;–Time Burgers equations C∂βu/∂tβ−(u·∇)u+(−Δ)u=0; etc.