Finite-time blow-up in a two-species chemotaxis-competition model with single production

Abstract

. This paper is concerned with blow-up of solutions to a two-species chemotaxis-competition model with production from only one species. In previous papers there are a lot of studies on boundedness for a two-species chemotaxis-competition model with productions from both two species. On the other hand, finite-time blow-up was recently obtained under smallness conditions for competitive effects. Now, in the biological view, the production term seems to promote blow-up phenomena; this implies that the lack of the production term makes the solution likely to be bounded. Thus, it is expected that there exists a solution of the system with single production such that the species which does not produce the chemical substance remains bounded, whereas the other species blows up. The purpose of this paper is to prove that this conjecture is true.