We consider a nonlinear parabolic equation with exponential nonlinearity which contains a nonlocal multiplication factor. This equation models an aggregation of cells via interaction with a chemical substance. We concentrate on two dimensional geometry, where a critical population number exists beyond which there are no regular steady states. The analysis involves a comparison theorem for linear parabolic equations with L1 sources, as well as a critical Sobolev inequality
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Gershon Wolansky (1997) studied this question.
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