Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity

We consider the quasilinear parabolic–parabolic Keller–Segel system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω⊂RnΩ⊂Rn with n⩾1n⩾1.It is proved that if S(u)D(u)⩽cuα with α<2n and some constant c>0c>0 for all u>1u>1, then the classical solutions to the above system are uniformly-in-time bounded, provided that D(u)D(u) satisfies some technical conditions such as algebraic upper and lower growth (resp. decay) estimates as u→∞u→∞. This boundedness result is optimal according to a recent result by the second author (Winkler, 2010 [27]), which says that if S(u)D(u)⩾cuα for u>1u>1 with c>0c>0 and some α>2n, n⩾2n⩾2, then for each mass M>0M>0 there exist blow-up solutions with mass ∫Ωu0=M∫Ωu0=M.In addition, this paper also proves a general boundedness result for quasilinear non-uniformly parabolic equations by modifying the iterative technique of Moser–Alikakos (Alikakos, 1979 [1]).