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Stability property of the two-dimensional Keller-Segel model. (English) Zbl 1184.35153
The connection between the parabolic-elliptic Keller-Segel (peKS) chemotaxis system \[ \partial_t u = \nabla\cdot (\nabla u - u \nabla v)\,, \quad -\Delta v = u\,, \quad (t,x)\in (0,\infty)\times\mathbb{R}^2\,, \] and the parabolic-parabolic Keller-Segel (ppKS) chemotaxis system \[ \partial_t u^\varepsilon = \nabla\cdot (\nabla u^\varepsilon - u^\varepsilon \nabla v^\varepsilon)\,, \quad \varepsilon \partial_t v^\varepsilon -\Delta v^\varepsilon = u^\varepsilon\,, \quad (t,x)\in (0,\infty)\times{\mathbb R}^2\,, \] as \(\varepsilon\to 0\) is investigated. More precisely, assume that the initial data \(u_0\in {\mathcal PM}^0\) and \(v_0\in {\mathcal PM}^2\) where \[ {\mathcal PM}^\alpha= \left\{ w \in {\mathcal S}'({\mathbb R}^2)\;:\;\|w\|_{{\mathcal PM}^\alpha} = \text{ ess sup}_{\xi\in {\mathbb R}^2} |\xi|^\alpha |\widehat{w}(\xi)| < \infty \right\} , \] where \(\widehat{w}\) denotes the Fourier transform of \(w\). Let \(\alpha\in (1/2,1)\). It is shown that, if \(\|u_0\|_{{\mathcal PM}^0}\), \(\|v_0\|_{{\mathcal PM}^2}\), and \(\varepsilon>0\) are sufficiently small, then there is a global solution \((u,v)\) to (peKS) and global solutions \((u^\varepsilon,v^\varepsilon)\) to (ppKS) such that \[ \lim_{\varepsilon\to 0} \sup_{t\geq 0}{\left\{ t^{\alpha/2} \|u(t) - u^\varepsilon(t)\|_{{\mathcal PM}^\alpha} \right\}} = 0 . \]

35K45 Initial value problems for second-order parabolic systems
35B20 Perturbations in context of PDEs
92C17 Cell movement (chemotaxis, etc.)
35B25 Singular perturbations in context of PDEs
35K58 Semilinear parabolic equations