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Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis. (English) Zbl 1458.35222

The author studies a predator-prey reaction-diffusion system with both predator- and prey- taxis, in a bounded spatial domain of dimension up to 3. Global existence is established, when the initial data is close enough to a stable steady state, as well as the convergence rate (either exponential or algebraic) toward this stable steady state. The main ideas of the proof are nicely laid out in the introduction; they rely on a suitable linear combination of certain functionals to cancel out problematic terms. This takes advantage of the structure of the system, especially the fact that coupling terms (taxis and reactions) have opposed signs in both equations.

MSC:

35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
92D25 Population dynamics (general)
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