Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K55, 76X05

Retrieve articles in all journals with MSC: 35K55, 76X05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0797051-0
PII: S 0002-9947(1985)0797051-0
Keywords: Cauchy problem, nonlinear diffusion, initial-value problem, regularizing effects
Article copyright: © Copyright 1985 American Mathematical Society