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A system of reaction diffusion equations arising in the theory of reinforced random walks. (English) Zbl 0874.35047

Summary: We investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks. We show that under some circumstances, finite-time blow-up of solutions is possible. In other circumstances, the solutions will decay to a spatially constant solution (collapse). We also give some intuitive arguments, which demonstrate the possibility of the existence of aggregation (piecewise constant) solutions.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35M10 PDEs of mixed type
35R25 Ill-posed problems for PDEs
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