On the heat diffusion starting with degeneracy

In this article we study the instant smoothing property of the heat diffusion that starts with degeneracy: u_t(t,x)=t^αΔu+f(t,x),t∈(0,T),x∈R^d;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.