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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Cauchy problem for $ u\sb t=\Delta u\sp m$ when $ 0<m<1$

Authors: Miguel A. Herrero and Michel Pierre
Journal: Trans. Amer. Math. Soc. 291 (1985), 145-158
MSC: Primary 35K55; Secondary 76X05
MathSciNet review: 797051
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Abstract: This paper deals with the Cauchy problem for the nonlinear diffusion equation $ \partial u/\partial t - \Delta (u\vert u{\vert^{m - 1}}) = 0$ on $ (0,\infty ) \times {{\mathbf{R}}^N},u(0, \cdot ) = {u_0}$ when $ 0 < m < 1$ (fast diffusion case). We prove that there exists a global time solution for any locally integrable function $ {u_0}$: hence, no growth condition at infinity for $ {u_0}$ is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and $ L_{\operatorname{loc} }^\infty $-regularizing effects are also examined when $ m \in (\max \{ (N - 2)/N,0\} ,1)$.

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Keywords: Cauchy problem, nonlinear diffusion, initial-value problem, regularizing effects
Article copyright: © Copyright 1985 American Mathematical Society

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