ULTRA-CONTRACTIVITY FOR KELLER-SEGEL MODEL WITH DIFFUSION EXPONENT m > 1-2/d

This paper establishes the hyper-contractivity in L-infinity(R-d) (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 vertical bar vertical bar U-0 vertical bar vertical bar(d)((2-m))(/2) < C-d,C-m where C-d,C-m is a universal constant, then for any t > 0, vertical bar vertical bar u(.,t)vertical bar vertical bar(L)infinity (R-d) is bounded and decays as t goes to infinity. For the subcritical case m > 2-2/d, the solution u(.,t) is an element of L-infinity (R-d) with any initial data U-0 is an element of L-+(1)(R-d) for any positive time.