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where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion. Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13, 14]. The original model was considered in 2D,

Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ be a standard slit domain where $\mathbb H$ is the upper half plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $\HH$. Given a Jordan arc $\gamma\subset D$ starting at $\partial {\mathbb H},$ let $g_t$ be the unique conformal map from $D\setminus \gamma[0,t]$ onto a standard slit domain $D_t$ satisfying the hydrodynamic normalization. We prove that $g_t$ satisfies an ODE with the kernel on its righthand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$, generalizing the chordal Loewner equation for the simply connected domain $D={\mathbb H}.$ Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in $t$. We establish the differentiability of $g_t$ in $t$ to make the equation a genuine ODE. To this end, we first derive the continuity of $g_t(z)$ in $t$ with a certain uniformity in $z$ from a probabilistic expression of $\Im g_t(z)$ in terms of the BMD for $D$, which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability.

Kadison and Kastler introduced a natural metric on the collection of all$C$*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear$C$*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.