Global Boundedness of Solutions to a Quasilinear Chemotaxis System with Nonlocal Nonlinear Reaction

This work studies a class of chemotaxis systems generalizing the prototype {u(t) = d & nabla; middot ((1 + u)(m-1)& nabla;u) - chi & nabla; middot (u(1 + u)(sigma-2)& nabla; v) + mu u(alpha)(1 - u(beta)),0 = delta v - v + u, with nonnegative initial data under zero-flux boundary conditions in a smooth bounded domain omega subset of R-N (N >= 1), where d, m, chi, mu > 0, sigma >= 1, and alpha,beta > 1. In this paper, it is rigorously proved that a global classical solution exists under the condition sigma + N/2 (sigma - m) - beta < alpha < m + 2/N beta. Moreover, the borderline case that alpha = sigma + N/2 (sigma - m) - beta is also taken into account and it is shown that a global classical solution exists when mu is suitably large.

Automated summary

This work studies a generalization of a chemotaxis system and proves the existence of global classical solutions under certain conditions. It also takes into account a borderline case and shows that a global classical solution exists with a suitably large parameter.