Global generalized solutions for a two-species chemotaxis system with tensor-valued sensitivity and logistic source

Abstract

This paper is devoted to investigating a two-species chemotaxis system with tensor-valued sensitivity and logistic source. It is mainly concerned with the global existence of generalized solutions. More precisely, when the logistic-type is weaker than the usual quadratic case, we prove, for any sufficiently smooth initial data, the associated no-flux/no-flux/no-flux problem possesses at least one globally defined solution in an appropriate generalized sense. In addition, we also underline that the similar results have been established for the corresponding system with constant sensitivity in a well-known result if the degradation is quadratic. Our results show that logistic source can rule out blow-up for such kind of models. On the other hand, due to the introduction of tensor value sensitivity, we neither need to impose any smallness assumption on the initial data here, nor require any restriction on the spatial dimension which generalizes and improves the partial of the result in the literature.