Boundedness in a chemotaxis model with exponentially decaying diffusivity and consumption of chemoattractant

It is proved that if D and S are sufficiently smooth nonnegative functions on [0,∞) satisfying K0e−β−s≤D(s)≤K1e−β+sfor alls≥0with some K0>0, K1>0, β−≥β+ and β+>0, then under the condition that S(s)D(s)≤K2sαfor alls≥0with some K2>0 and α≥0, for the initial data (u0,v0) are sufficiently regular satisfying u0≥0 and v0>0, the classical solutions to the system are uniformly-in-time bounded.