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

<jats:title>Abstract</jats:title><jats:p>This work studies the two‐species Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by <jats:disp-formula> </jats:disp-formula>with positive parameters and , and nonnegative constants and . 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 <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#plms12276-disp-0001" /> seem unsolved so far.</jats:p><jats:p>In the present paper it is <jats:italic>inter alia</jats:italic> shown that if and is a bounded convex domain with smooth boundary, then whenever and are nonnegative, the associated Neumann initial‐boundary value problem for <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#plms12276-disp-0001" /> possesses a global classical solution which in fact is bounded in the sense that <jats:disp-formula> </jats:disp-formula>Moreover, the asymptotic behavior of arbitrary nonnegative solutions enjoying the boundedness property is studied in the general situation when is arbitrary and no longer necessarily convex. If , then in both cases and , an explicit smallness condition on is identified as sufficient for stabilization of any nontrivial solutions toward a corresponding unique nontrivial spatially homogeneous steady state. If and , 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 <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#plms12276-disp-0001" />.</jats:p>