Global boundedness to a parabolic-parabolic chemotaxis model with nonlinear diffusion and singular sensitivity

Abstract

This article deals with the parabolic-parabolic chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ φ ( v ) ) + f ( u ) , x ∈ Ω , t > 0 , v t = △ v − v + u , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R n ( n ≥ 1 ) with smooth boundary conditions, D , S ∈ C 2 ( [ 0 , + ∞ ) ) nonnegative, with D ( u ) = a 0 ( u + 1 ) − α for a 0 > 0 and α < 0 , 0 ≤ S ( u ) ≤ b 0 ( u + 1 ) β for b 0 > 0 , β ∈ R , and where the singular sensitivity satisfies 0 < φ ′ ( v ) ≤ χ v k for χ > 0 , k ≥ 1 . In addition, f : R → R is a smooth function satisfying f ( s ) ≡ 0 or generalizing the logistic source f ( s ) = r s − μ s m for all s ≥ 0 with r ∈ R , μ > 0 , and m > 1 . It is shown that for the case without a growth source, if 2 β − α < 2 , the corresponding system possesses a globally bounded classical solution. For the case with a logistic source, if 2 β + α < 2 and n = 1 or n ≥ 2 with m > 2 β + 1 , the corresponding system has a globally classical solution.