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On the heat diffusion starting with degeneracy

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Abstract In this article we study the instant smoothing property of the heat diffusion that starts with degeneracy: u t ( t , x ) = t α Δ u… Click to show full abstract

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

Keywords: heat; diffusion starting; starting degeneracy; heat diffusion

Journal Title: Journal of Differential Equations
Year Published: 2017

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