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The taxis-type migration-consumption model accounting for signal-dependentmotilities, as given by \[  u_t = \Delta \big(u\phi(v)\big),  v_t = \Delta v-uv,  \qquad (*) \] is considered for suitably smooth functions $\phi:[0,\infty)\toR$ which are such that $\phi>0$ on $(0,\infty)$, but that in addition$\phi(0)=0$ with $\phi'(0)>0$.  In order to appropriately cope with the diffusion degeneracies therebyincluded, this study separately examines the Neumann problem for the linearequation \[  V_t = \Delta V + \nabla\cdot \big( a(x,t)V\big) + b(x,t)V \] and establishesa statement on how pointwise positive lower bounds for nonnegative solutionsdepend on the supremum and the mass of the initial data, and on integrabilityfeatures of $a$ and $b$.  This is thereafter used as a key tool in the derivation of a result on globalexistence of solutions to (*), smooth and classical for positive times, underthe mere assumption that the suitably regular initial data be nonnegative inboth components. Apart from that, these solutions are seen to stabilize towardsome equilibrium, and as a qualitative effect genuinely due to degeneracy indiffusion, a criterion on initial smallness of the second component isidentified as sufficient for this limit state to be spatially nonconstant.

A quantitative strong parabolic maximum principle and application to a  taxis-type migration-consumption model involving signal-dependent degenerate  diffusion