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The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations
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- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 3 / 01 September 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 623-631
- Print publication:
- 01 September 2012
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- Article
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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
$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$
