A strongly degenerate migration-consumption model in domains of arbitrary dimension

2023/12/19 by Winkler, Michael · 1 citation
#35K51 #35K57 #35K65 #35Q92 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.12409

Abstract

In a smoothly bounded convex domain Ω⊂ Rⁿ with n≥ 1, a no-flux initial-boundary value problem for \ u_t=Δ(uϕ(v)), v_t=Δv-uv, . is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by ϕ(ξ)=ξ^α, ξ∈ [0,ξ₀]. By means of separate approaches, it is shown that in both cases α∈ (0,1) and α∈ [1,2] some global weak solutions exist which, inter alia, satisfy C(T):= \rm esssup _{t∈ (0,T)} ∫_Ωu(⋅,t)ln u(⋅,t) < ∞ for all T>0, with sup_T>0 C(T)<∞ if α∈ [1,2].

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