Global boundedness of solutions to a quasilinear chemotaxis system with nonlocal nonlinear reaction (Q2682360)

The authors study a quasilinear parabolic-elliptic chemotaxis system \[ u_t=d\nabla\cdot((1+u)^{m-1}\nabla u)-\chi\nabla\cdot((u(1+u)^{\sigma-2}\nabla u)+\mu u^\alpha\left(1-\int_\Omega u^\beta\right), \] \[ \Delta v-v+u=0, \] with no-flux boundary conditions in a bounded domain of \(\mathbb R^N\). The nonlocal term may model the proliferation of the population that depends on an integral functional of the population density, i.e. depends in a nonlocal way on the population. The main result is the existence of a global-in-time classical solution assuming the condition \[ \sigma+\frac{N}{2}(\sigma-m)-\beta<\alpha<m+\frac{2}{N}\beta. \]