Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source. (December 2017) - Wang, Yifu; Liu, Ji
Abstract: In this paper, we consider the quasilinear chemotaxis system ( ⋆ ) u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) + f ( u ), x ∈ Ω, t > 0, v t = Δ v − v + u, x ∈ Ω, t > 0 in a bounded domain Ω ⊂ R n ( n ≥ 2 ) under zero-flux boundary conditions, where the nonlinearities D, S ∈ C 2 ( [ 0, ∞ ) ) are supposed to generalize the prototypes D ( u ) = C D ( u + 1 ) m − 1 and S ( u ) = C S u ( u + 1 ) q − 1 with C D, C S > 0 and m, q ∈ R, and f ∈ C 1 ( [ 0, ∞ ) ) satisfies f ( u ) ≤ r − b u γ with r ≥ 0, b > 0 and γ > 1 . It is shown that if q < max m + 2 n − 1, m + γ 2 − n − 1 n for 1 < γ ≤ n + 2 n, max m + 2 γ n + 2 − 1, m 2 + γ ( n + 4 ) 2 ( n + 2 ) − 1 for n + 2 n < γ < n + 2 2, max m + 2 n − n + 2 n γ, m 2 + γ ( n + 4 ) 2 ( n + 2 ) − 1 for n + 2 2 ≤ γ < n + 2, max m + 1 n, m + γ 2 for γ ≥ n + 2, then(⋆) has a unique globally bounded classical solution.,