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Journal Article

# The Cauchy Problem for ut = Δ um When $0 < m < 1$

Miguel A. Herrero and Michel Pierre
Transactions of the American Mathematical Society
Vol. 291, No. 1 (Sep., 1985), pp. 145-158
DOI: 10.2307/1999900
Stable URL: http://www.jstor.org/stable/1999900
Page Count: 14

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## Abstract

This paper deals with the Cauchy problem for the nonlinear diffusion equation ∂ u/∂ t - Δ(u|u|m-1) = 0 on (0, ∞) × RN, u (0, ·) = u0 when $0 < m < 1$ (fast diffusion case). We prove that there exists a global time solution for any locally integrable function u0: hence, no growth condition at infinity for u0 is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and $L^\infty_{\operatorname{loc}}$ -regularizing effects are also examined when m ∈ (max {(N - 2)/N,0}, 1).

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