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.
- Degenerate parabolic equation
- Instant smoothing property
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