Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source

In this paper, we are concerned with a general class of quasilinear parabolic-parabolic chemotaxis systems with/without growth source, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn with n≥2. It is recently known that blowup is possible even in the presence of superlinear growth restrictions. Here, we derive new and interesting characterizations on the growth versus the boundedness. We show that the hard task of proving the L∞-boundedness of the cell density can be reduced to proving its Lr-boundedness. In other words, we show that the Lr-boundedness of the cell density can successfully guarantee its L∞-boundedness and hence its global boundedness, where r=n+ϵ or n2+ϵ depending on whether the growth restriction is essentially linear (including no growth) or superlinear. Hence, a blowup solution also blows up in Lp-norm for any suitably large p. More detailed information on how the growth source affects the boundedness of the solution is derived. These results reveal deep understandings of blowup mechanism for chemotaxis models. Then we use these criteria to establish uniform boundedness and hence global existence of the underlying models: logistic source in 2-D, cubic source as initially proposed by Mimura and Tsujikawa in 3-D, [(n−1)+ϵ]st source in n-D with n≥4. As a consequence, in a chemotaxis-growth model, blowup is impossible if the growth effect is suitably strong. Finally, we underline that our results remove the commonly assumed convexity on the domain Ω.