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Boundedness vs. blow-up in a chemotaxis system. (English) Zbl 1085.35065

Authors’ abstract: 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.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92C17 Cell movement (chemotaxis, etc.)
35B40 Asymptotic behavior of solutions to PDEs
35B33 Critical exponents in context of PDEs
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