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

TL;DR

This paper develops a quantitative maximum principle for degenerate diffusion equations and applies it to prove global existence, stability, and nonconstant equilibrium states in a signal-dependent taxis migration model.

Contribution

It introduces a new strong maximum principle for degenerate diffusion equations and applies it to establish global solutions and their qualitative behavior in a complex migration model.

Findings

Established pointwise positive lower bounds for solutions of degenerate linear equations.

Proved global existence and smoothness of solutions to the migration model.

Identified conditions for solutions to stabilize to nonconstant equilibria.

Abstract

The taxis-type migration-consumption model accounting for signal-dependent motilities, as given by \[ u_t = \Delta \big(u\phi(v)\big), v_t = \Delta v-uv, \qquad (*) \] is considered for suitably smooth functions $\phi:[0,\infty)\to R$ 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 thereby included, this study separately examines the Neumann problem for the linear equation \[ V_t = \Delta V + \nabla\cdot \big( a(x,t)V\big) + b(x,t)V \] and establishes a statement on how pointwise positive lower bounds for nonnegative solutions depend on the supremum and the mass of the initial data, and on integrability features of $a$ and $b$. This is thereafter used as a key tool in the derivation of a result on global existence of solutions to (*), smooth and classical for positive times, under the mere assumption that the suitably regular initial data be nonnegative in both components. Apart from that, these solutions are seen to stabilize toward some equilibrium, and as a qualitative effect genuinely due to degeneracy in diffusion, a criterion on initial smallness of the second component is identified as sufficient for this limit state to be spatially nonconstant.