Boundedness and stabilization in a population model with cross‐diffusion for one species

This work studies the two‐species Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by ⋆u t = Δ [ ( d 1 + a 11 u + a 12 v ) u ] + μ 1 u ( 1 − u − a 1 v ) ,v t = Δ [ ( d 2 + a 22 v ) v ] + μ 2 v ( 1 − v − a 2 u ) ,with positive parametersd 1 , d 2and a 11 , and nonnegative constantsa 12 , a 22 , μ 1 , μ 2 , a 1and a 2 . Beyond some statements on global existence, the literature apparently provides only few results on qualitative behavior of solutions; in particular, questions related to boundedness as well as to large time asymptotics in seem unsolved so far. In the present paper it is inter alia shown that if n ⩽ 9 and Ω ⊂ R nis a bounded convex domain with smooth boundary, then wheneveru 0 ∈ W 1 , ∞( Ω )andv 0 ∈ W 1 , ∞( Ω )are nonnegative, the associated Neumann initial‐boundary value problem for possesses a global classical solution which in fact is bounded in the sense that ⋆⋆ u ∈ L ∞ ( Ω × ( 0 , ∞ ) )and v ∈ L ∞ ( ( 0 , ∞ ) ; W 1 , p( Ω ) )for all p > n . Moreover, the asymptotic behavior of arbitrary nonnegative solutions enjoying the boundedness property is studied in the general situation when n ⩾ 1 is arbitrary and Ω no longer necessarily convex. Ifa 1 ∈ ( 0 , 1 ) , then in both casesa 2 > 1 anda 2 ∈ ( 0 , 1 ) , an explicit smallness condition on a 12 is identified as sufficient for stabilization of any nontrivial solutions toward a corresponding unique nontrivial spatially homogeneous steady state. Ifa 1 ⩾ 1 anda 2 ∈ ( 0 , 1 ) , then without any further assumption all nonzero solutions are seen to approach the equilibrium (0,1). As a by‐product, this particularly improves previous knowledge on nonexistence of nonconstant equilibria of .