Continuous solutions for a two-dimensional cross-diffusion problem involving doubly degenerate diffusion and logistic proliferation

Abstract

Emerging from the modeling of intricate structures formed by bacterial motion in nutrient-deficient environments, the doubly degenerate nutrient taxis system [Formula: see text] is considered in smoothly bounded domains [Formula: see text]. In contrast to frameworks devoid of logistic growth where only small-data global solutions have so far been found in the literature, quadratic degradation in the logistic growth is shown to markedly amplify the overall dissipative effects to the extent that, for arbitrary smooth initial data, an associated no-flux type initial-boundary problem possesses global continuous weak solutions. Central to our findings is the identification of an advantageous dissipative structure subtly embedded in the system, the exploitation of which leads to the establishment of a positive lower bound for nutrients levels within any finite time interval, thereby ruling out the appearance of cross-degeneracies within such timeframes.