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The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations

Published online by Cambridge University Press:  20 November 2018

Jiaqing Pan
Jiaqing Pan*
Affiliation:
Institute of Mathematics, Jimei University, Xiamen, 361021, P. R. Chinae-mail: [email protected]
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Abstract

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In this paper, we consider the Cauchy problem

$$\left\{ \begin{align} & {{u}_{t}}=\Delta ({{u}^{m}}),\,\,\,\,\,x\in {{\mathbb{R}}^{N}},t>0,N\ge 3, \\ & u(x,0)={{u}_{0}}(x),\,\,\,\,\,x\in {{\mathbb{R}}^{N}}. \\ \end{align} \right.$$

We will prove that

(i) for ${{m}_{c}}\,<\,m,\,{{m}_{0}}\,<\,1,\,\left| u(x,\,t,m)-u(x,\,t,{{m}_{0}}) \right|\,\to \,0$ as $m\,\to \,{{m}_{0}}$ uniformly on every compact subset of ${{\mathbb{R}}^{N}}\,\times \,{{\mathbb{R}}^{+}}$, where ${{m}_{c}}\,=\,\frac{{{(N-2)}_{+}}}{N}$;

(ii) there is a ${{C}^{*}}$ that explicitly depends on $m$ such that

$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$

Keywords

35K0535K1035K15fast diffusion equationsCauchy problemcontinuous dependence on nonlinearity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 623 - 631
Copyright
Copyright © Canadian Mathematical Society 2012

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