Global Boundedness and Mass Persistence in a Multi-Species Chemotaxis System

This research paper concerns with the population dynamics of a multi-species and multi-chemicals chemotaxis system characterized by a parabolic-parabolic-elliptic-elliptic structure under no-flux boundary conditions in a smooth bounded domain. This research study examines the global existence, global boundedness, and persistence of mass of solutions of the system mentioned above. In all spatial dimensional settings, we first demonstrate the global Lᵖ-boundedness of solutions under some explicit parameter conditions that notably exclude any dependence on the dimensionality. Then, it has been establihed that the global existence and boundedness of positive solutions are implied by Lᵖ-bounds of solutions under the exact same hypotheses. In addition to these ones, we prove that any globally bounded classical solution eventually persist as a whole under the same conditions. The results obtained in this study contribute to a more profound theoretical understanding of chemotaxis models in multi-species and multi-chemical environments. In order to establish the qualitative properties of chemotaxis model mentioned in the above, some advanced mathematical techniques and strategies has been developed.