Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities

The chemotaxis system document arrayl\ array{l uₜ = (D (u) u) - (uS (x, u, v) v), \\ vₜ = v -uv, array. array document is considered in a bounded domain document Rⁿ document, document n 2 document, with smooth boundary. It is shown that if document D: [0, ) [0, ) document and document S: [0, ) (0, ) R^n n document are suitably smooth functions satisfying document arraylD (u) kD u^m-1 for\; all\; u 0 array document and document arrayl|S (x, u, v) | S₀ (v) v^ for\; all\; (x, u, v) \; (0, ) ² array document with some document arraylm3n-22n and\; [0, 1), array document and with some document kD0 document and nondecreasing document S₀: (0, ) (0, ) document, then for all suitably regular initial data a corresponding no-flux type initial-boundary value problem admits a global bounded weak solution which actually is smooth and classical if document D (0) 0 document.