Publications [#246858] of Jian-Guo Liu

Papers Published

1. Bian, S; Liu, JG; Zou, C, Ultra-contractivity for keller-segel model with diffusion exponent m > 1-2/d, Kinetic and Related Models, vol. 7 no. 1 (March, 2014), pp. 9-28, American Institute of Mathematical Sciences (AIMS), ISSN 1937-5093 [doi]
   (last updated on 2024/04/22)

   Abstract:
   This paper establishes the hyper-contractivity in L∞(ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent m > 1-2/d. The results show that for the super- critical and critical case 1-2/d < m ≤ 2-2/d, if ∥U0∥d(2-m)/2 < Cd, m where Cd, m is a universal constant, then for any t > 0 ∥u(.,t)∥L∞(ℝd) is bounded and decays as t goes to infinity. For the subcritical case m > 2-2/d, the solution u(.,t)∈ L∞(ℝd) with any initial data U0 ∈ L1+(ℝd) for any positive time.