Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two

Abstract

This paper deals with the parabolic-elliptic Keller–Segel system with weakly singular sensitivity and logistic source under the homogeneous Neumann boundary: u t = Δ u − χ ∇ ⋅ ( u v α ∇ v ) + r u − μ u 2 , 0 = Δ v − v + u in a smooth bounded domain Ω ⊂ R 2 , with χ , α , μ > 0 , r ∈ R . It is proved that the system possesses a globally bounded classical solution for α ∈ ( 0 , 1 ) with µ > 0 suitably large, without establishing the uniformly positive bound for v from below. In addition, we give the explicit expression of the upper bound for solution u with respect to the parameters χ , α , r , μ via a recursive argument on α. This concludes that weakly singular sensitivity benefits to obtain the global boundedness of classical solution in dimension two.