# zbMATH — the first resource for mathematics

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 $\lim_{\varepsilon\to 0} \sup_{t\geq 0}{\left\{ t^{\alpha/2} \|u(t) - u^\varepsilon(t)\|_{{\mathcal PM}^\alpha} \right\}} = 0 .$

##### MSC:
 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