On $\varepsilon$-Regularity Theorem and Asymptotic Behaviors of Solutions for Keller–Segel Systems

Abstract

We deal with the equation (KS)$_m$ for the critical case of $q = m + 2/N$ with $N \ge 3$, $m > 1$, $q \ge 2$: $\partial_t u = \Delta u^m - \nabla \cdot (u^{q-1} \nabla v)$, $x \in \mathbb{R}^N$, $t>0$; $0 = \Delta v - \gamma v + u$, $x \in \mathbb{R}^N$, $t>0$; $u(x,0) = u_0(x)$, $\tau v(x,0) = \tau v_0(x)$, $x \in \mathbb{R}^N$. Based on a $\varepsilon$-regularity theorem in [Y. Sugiyama, Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type, submitted], we first show that the set $S_u$ of blow-up points of the weak solution u has at most the zero-Hausdorff dimension if $u \in C_w([0,T]; L^1(\mathbb{R}^N))$. Next, we give various conditions on the weak solution u so that the set $S_u$ consists of finitely many points. Furthermore, we obtain an explicit constant for $\varepsilon$ in such a way that if the local concentration of mass around some point $x \in S_u$ is less than $\varepsilon$, then u is in fact locally bounded around x, which may be regarded as a removable singularity theorem. Simultaneously, we shall show that the solution u in $C([0,T]; L^1(\mathbb{R}^N))$ can be continued beyond $t=T$, which gives an extension criterion in the scaling invariant class associated with (KS)$_m$.