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2006 Extended solutions for general fast diffusion equations with optimal measure data
Emmanuel Chasseigne, Juan Luis Vazquez
Adv. Differential Equations 11(6): 627-646 (2006).

Abstract

We study the theory of existence and uniqueness of nonlinear diffusion equations of the form \[ u_{t}=\Delta \varphi(u)\quad\text{in}\quad\mathbb{R}^{N} \times(0,\infty), \] where $\varphi:{{\mathbb{R}^N}}_+\to{{\mathbb{R}^N}}_+$ is a continuous and increasing function. We focus on the fast diffusion type by imposing the growth condition : $$ \qquad {\displaystyle} m_1\le \dfrac{s\varphi'(s)}{\varphi(s)} \le m_2, $$ for some constants $0 < m_1\le m_2 < 1$ and all $s>0$. Moreover, $m_1>(N-2)/N$. Existence is obtained for an optimal class of initial data, namely, for any nonnegative Borel measure (not necessarily a locally finite measure). Uniqueness is proven for concave $\varphi$'s. The results extend to the general equation the optimal theory with measure-valued data now available for the equation with power functions $\varphi(s)=s^{m}$. The asymptotic behaviour is also studied.

Citation

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Emmanuel Chasseigne. Juan Luis Vazquez. "Extended solutions for general fast diffusion equations with optimal measure data." Adv. Differential Equations 11 (6) 627 - 646, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1173.35561
MathSciNet: MR2238022

Subjects:
Primary: 35K55
Secondary: 35K65

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.11 • No. 6 • 2006
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