Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis

This paper studies a reaction–diffusion system of a predator–prey model with Holling type II functional response and prey-taxis, proposed by Ainseba et al. (2008), where the prey-taxis means a direct movement of the predator in response to a variation of the prey (which results in the aggregation of the predator). The global existence of classical solutions was established by Tao (2010). In this paper we prove furthermore that the global classical solutions are globally bounded, by means of the Gagliardo–Nirenberg inequality, the Lp−LqLp−Lq estimates for the Neumann heat semigroup, and the LpLp estimates with Moser’s iteration of parabolic equations.