On the heat diffusion starting with degeneracy

In this article we study the instant smoothing property of the heat diffusion that starts with degeneracy:ut(t,x)=tαΔu+f(t,x),t∈(0,T),x∈Rd;u(0,x)=u0(x), where α∈(−1,∞). We provide the existence and uniqueness result in an appropriate Sobolev space setting. For a fixed f the regularity improvement in Sobolev regularity from u0 to u changes continuously along α. In particular, the larger α>0, the smaller the improvement is. Moreover, we study a regularity relation between f and u near time t=0 as α varies.