Boundedness to a quasilinear chemotaxis-consumption system with singular sensitivity in dimension one (Q2239282)

A one-dimensional chemotaxis system \[u_t=(D(u)u_x)_x-\chi\left(\frac{S(u)}{v}v_x\right)_x,\] \[v_t=v_{xx}-uv,\] with power-bounded coefficients \(D\) and \(S\) is considered on a bounded interval. Sufficient conditions for uniform boundedness of solutions are derived which improve, in some cases, known results.