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

Pan, Jiaqing

Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 623-631

Résumé

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

Détail

DOI : 10.4153/CMB-2011-085-9

Mots-clés : 35K05, 35K10, 35K15, fast diffusion equations, Cauchy problem, continuous dependence on nonlinearity