L~?-measure Criteria For Boundedness In A Quasilinear Parabolic-elliptic Keller-segel System With Supercritical Sensitivity

This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity: u_t=?·?D?u??u?-??S?u??v?,0 = ?v-v + u in ? ×?0,T?,where bounded domain ? ? Rⁿ,n ? 2,subject to the non-flux boundary conditions,D?u?????u + 1?^-q,S?u????u?u + 1?^?-q-1 with q ? R,? ?2/n.Concerning the suitable measure for initial data to be determined,it is proved that the problem possesses a unique globally bounded solution for ? ?2/n whenever ||u₀||L^? is sufficiently small with ? >n?/2+q if 0 < q <2/n,or ? >n?/2 if q < 0.