This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as 1663

We consider the classical parabolic-parabolic Keller-Segel systemunder homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n .It is proved that in space dimension n 3, for each q > n 2 and p > n one can find ε 0 > 0 such that if the initial data (u 0 , v 0 ) satisfy u 0 L q (Ω) < ε and ∇ v 0 L p (Ω) < ε then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations. In particular, (u, v) approaches the steady state (m, m) as t → ∞, where m is the total mass m := Ω u 0 of the population. Moreover, we shall show that if Ω is a ball then for arbitrary prescribed m > 0 there exist unbounded solutions emanating from initial data (u 0 , v 0 ) having total mass Ω u 0 = m.

We determine the critical blow-up exponent for a Keller-Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.

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We consider the quasilinear parabolic-parabolic Keller-Segel systemunder homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n with n ≥ 2. It is proved that if S(u) D(u) ≤ cu α with α < 2 n and some constant c > 0 for all u > 1 and some further technical conditions are fulfilled, then the classical solutions to the above system are uniformly-in-time bounded. This boundedness result is optimal according to a recent result by the second author (Math. Meth. Appl. Sci. 33 (2010), 12-24), which says that if S(u) D(u) ≥ cu α for u > 1 with c > 0 and some α > 2 n , then for each mass M > 0 there exist blow-up solutions with mass Ω u 0 = M . In addition, this paper also proves a general boundedness result for quasilinear nonuniformly parabolic equations by modifying the iterative technique of Moser-Alikakos (Alikakos, Comm. PDE 4 (1979), 827-868).

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