Proliferation-dominated large time behaviour in a planar doubly degenerate nutrient taxis system

This study is concerned with nonnegative solutions of the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system with a logistic source, as given by {align*}\{{array}{ll}u_t = ∇· (uv∇ u) - ∇· (u^2 v∇ v)+ u - u^2,\\[1mm]v_t = Δ v -uv,{array} .{align*} in a smoothly bounded planar domain Ω with (u,v)|ₜ₌₀=(u₀,v₀) . It is shown that despite substantially weakened diffusion, the stabilizing effects of logistic-type cell kinetics may overbalance any heterogeneity-supporting tendency of cross-diffusion: Namely, it is seen that for all suitably smooth initial data satisfying v₀ > 0 in Ω̄ and {align*}∫_Ω ln u_0~ > ~-∞,{align*} a corresponding continuous weak solution, known to exist globally according to previous literature, has the property that as t→∞ , {align*} u(·,t) 1 in L^1(Ω) v(·,t)→ 0 in L^∞(Ω). {align*}