## Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis.(English)Zbl 1195.35171

Summary: This paper deals with a predator-prey model consisting of a $$2\times 2$$ reaction-diffusion-taxis system recently proposed by B. E. Ainseba, M. Bendahmane and A. Noussair [Nonlinear Anal., Real World Appl. 9, 2086–2105 (2005; Zbl 1156.35404)]. The central point of this system is that the spatial-temporal variations of the predators’ velocity are directed by the prey gradient. The global existence and uniqueness of classical solutions to this system are proved by the contraction mapping principle, together with $$L^p$$ estimates and Schauder estimates of parabolic equations. The crucial point of the proof is to deal with the prey-tactic term with a nonlinear tactic function.

### MSC:

 35K51 Initial-boundary value problems for second-order parabolic systems 35K58 Semilinear parabolic equations 92D25 Population dynamics (general) 35K57 Reaction-diffusion equations

Zbl 1156.35404
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### References:

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