Global solutions of a doubly tactic resource consumption model with logistic source

Abstract

We study a doubly tactic resource consumption model (ut = Δu − ∇ · (u∇w), vt = Δv − ∇ · (v∇u) + v(1 − vβ−1), wt = Δw − (u + v)w − w + r) in a smooth bounded domain Ω∈R2 with homogeneous Neumann boundary conditions, where r∈C1(Ω̄×[0,∞))∩L∞(Ω×(0,∞)) is a given non-negative function fulfilling ∫tt+1∫Ω|∇r|2<∞ for all t ≥ 0. It is shown that, first, if β > 2, then the corresponding Neumann initial-boundary problem admits a global bounded classical solution. Second, when β = 2, the Neumann initial-boundary problem admits a global generalized solution.