On the well-posedness via the JKO approach and a study of blow-up of solutions for a multispecies Keller-Segel chemotaxis system with no mass conservation

Abstract

In this study, we consider a multispecies chemotaxis system that includes birth or death rate terms, which means that there is no mass conservation of the populations. First, in the spirit of [52] and [18], we demonstrate the convergence of the JKO scheme (derived from the Optimal Transport theory) to an L∞-weak solution that is local in time. Recently, L∞ solutions have shown to be important to obtaining uniqueness results. Since the death rate case does not ensure the existence of global L∞ solutions for arbitrary initial data, we establish sufficient conditions that lead to the finite-time blow-up phenomenon and describe several stages at which this occurs. This part can be seen as a partial generalization of the blow-up results reported in [22]. Finally, we conduct some numerical simulations that explore solutions for initial data types not covered in the main convergence result.