Ultra-contractivity for keller-segel model with diffusion exponent m > 1-2/d
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.
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