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{ut=∇⋅(D(u)∇u)−∇⋅(uv∇v)in Ω×(0,∞)vt=Δv−uvin Ω×(0,∞)∂νu=∂νv=0in ∂Ω×(0,∞)u(⋅,0)=u0,v(⋅,0)=v0in Ω in smooth bounded domains Ω⊂RN, N≥2, for D(u)≥δum−1 with some δ>0, provided that m>1+N4.