A degenerate $p$-Laplacian Keller-Segel model

Jian-Guo Liu,Wenting Cong

doi:10.3934/krm.2016012

Abstract

This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.

Highlights

In this paper, we study the following p-Laplacian Keller-Segel model in d ≥ 3: ∂tu = ∇ · |∇u|p−2∇u− ∇ · (u∇v), x ∈ Rd, t > 0,−∆v = u, x ∈ Rd, t > 0, (1)u(x, 0) = u0(x), x ∈ Rd, where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion
We prove the local existence of weak solutions and a blow-up criterion for general L1 ∩ L∞ initial data
If u0 Lq(Rd) < Cd,p, where Cd,p is a universal constant depending on d and p, we will show that there exists a global weak solution

Summary

Introduction

If u0 Lq(Rd) < Cd,p, where Cd,p is a universal constant depending on d and p, we will show that there exists a global weak solution. We prove Theorem 3.1 which is concerning a priori estimates of weak solutions of (1). 4. The uniformly in time L∞ estimate of weak solutions.

Results

Conclusion