Application of the Moser-Trudinger inequality in the construction of global solutions to a strongly degenerate migration model

Abstract

A no-flux initial-boundary value problem for the cross-diffusion system ut = Δ(uφ(v)),vt = Δv - uv is considered in smoothly bounded domains ω ⊂ ℝn with n ≤ 2. It is shown that whenever φ C0([0,∞)) is positive on (0,∞) and such that lim infζ↘0φ(ζ) ζα > 0(⋆) for some α > 0, for all suitably regular positive initial data a global very weak solution, particularly preserving mass in its first component, can be constructed. This extends previous results which either concentrate on non-degenerate analogs, or are restricted to the special case α = 1. To appropriately cope with the considerably stronger cross-degeneracies thus allowed through (⋆) when α is large, in its core part the analysis relies on the use of the Moser-Trudinger inequality in controlling the respective diffusion rates φ(v) from below.