Global solutions to the higher-dimensional chemotaxis-consumption system

Abstract

This paper is devoted to a parabolic-parabolic chemotaxis-consumption PDE's system with singular sensitivity under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subseteq R^n$, $n\ge 2$. We consider a growth term of logistic type in the equation of “u” in the form $u^{\alpha }(1-{\int }_{\Omega }{u}^{\beta })$. We provide conditions to ensure global existence of solutions. As compared to previous mathematical studies, the novelty (also is difficulty) of this problem arises from the combination of the singular sensitivity and the nonlocal nonlinear reaction.