GLOBAL SOLVABILITY TO A SINGULAR CHEMOTAXIS-CONSUMPTION MODEL WITH FAST AND SLOW DIFFUSION AND LOGISTIC SOURCE

In this paper, we consider the following chemotaxis-consumption model with porous medium diffusion and singular sensitivity {u(t) = Delta u(m) - chi div(u/v del v) + mu u(1-u), vt - Delta v - u(r)v, in a bounded domain Omega subset of R-N (N >= 2) with zero-flux boundary conditions. It is shown that if r < 4/N+2, for arbitrary case of fast diffusion (m <= 1) and slow diffusion (m > 1), this problem admits a locally bounded global weak solution. It is worth mentioning that there are no smallness restrictions on the initial datum and chemotactic coefficient.