Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

We show the existence of locally bounded global solutions to the chemotaxis system $$u_t=\mathrm{\nabla }\cdot (D(u)\mathrm{\nabla }u)-\mathrm{\nabla }\cdot (\frac{u}{v}\mathrm{\nabla }v)$$ $$v_t=\mathrm{\Delta }v-uv$$ with homogeneous Neumann boundary conditions and suitably regular positive initial data in smooth bounded domains $\Omega \subset $\mathbb{R}$^N$,$N\geq2$,for$D(u)\geq \delta u^{m-1}$withsome$\delta>0$,providedthat$m>1+\frac N4$.