Small-density solutions in Keller–Segel systems involving rapidly decaying diffusivities

Abstract

In a smoothly bounded domain $\Omega \subset \mathbb {R}^n$, $n\ge 1$, the quasilinear Keller–Segel system

is considered under homogeneous no-flux boundary conditions. It is firstly shown that if D and S, besides belonging to $C^2([0,\infty ))$ with $S(0)=0$, merely satisfy

then for all $K>0$ there exists $\varepsilon _\star (K)\in (0,\frac{R}{2})$ such that whenever $0\le u_0\in W^{1,\infty }(\Omega )$ and $0\le v_0\in W^{1,\infty }(\Omega )$ satisfy

a corresponding initial value problem for ($\star$) admits a global bounded classical solution with $(u,v)|_{t=0}=(u_0,v_0)$. Secondly, a more restrictive condition on the initial data, inter alia requiring appropriate smallness of both $\Vert u_0\Vert _{L^\infty (\Omega )}$ and $\Vert v_0\Vert _{W^{1,\infty }(\Omega )}$, is identified as sufficient to ensure exponential stabilization of the correspondingly obtained solution toward the equilibrium $(\frac{1}{|\Omega |} \int _\Omega u_0, \frac{1}{|\Omega |}\int _\Omega u_0)$. As a technical ingredient of crucial importance for the derivation of explicit pointwise bounds for the respective first solution components, the analysis relies on a refinement of a Moser-type iterative argument which, formulated here in a general context of parabolic inequalities, provides some quantitative information about the dependence of $L^\infty$ estimates on bounds on the initial data and $L^1$ bounds.