A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity (Q2671207)

A fully parabolic chemotaxis system with nondegenerate diffusion (with the coefficient \(D(u)\ge (u+1)^\alpha\)), singular sensitivity term (\(\frac{u}{v}\nabla v\)) and logistic reaction term is considered in bounded domains of \(\mathbb R^n\). The author proves results on the global existence, boundedness of classical solutions, and their convergence to homogeneous steady states. The crucial assumption is that \(\alpha>1-\frac{4}{n+2}\), and suitable Lyapunov functionals are constructed to prove these properties.