TY - JOUR

T1 - Global Solutions near Homogeneous Steady States in a Multidimensional Population Model with Both Predator- and Prey-Taxis

AU - Fuest, Mario

N1 - Publisher Copyright: © 2020 Society for Industrial and Applied Mathematics.

PY - 2020/1

Y1 - 2020/1

N2 - We study the system () (equation presented) (inter alia) for D1,D2, χ 1, χ 2, λ 1, λ 2, μ 1, μ 2, a1, a2 > 0 in smooth, bounded domains Ω ⊂ Rn, n ∈{ 1, 2, 3} . Without any further restrictions on these parameters, we prove that there exists a constant stable steady state (u , v ) ∈ [0,∞ )2, meaning that there is ϵ > 0 such that if u0, v0 ∈ W2,2(Ω) are nonnegative with ∂ ν u0 = ∂ ν v0 = 0 in the sense of traces and | u0 u | W2,2(Ω)+| v0 v | W2,2(Ω) < ϵ , then there exists a global classical solution (u, v) of (*) with initial data u0, v0 converging to (u , v ) in W2,2(Ω). Moreover, the convergence rate is exponential, except for the case λ 2μ 1 = λ 1a2, where it is is only algebraical. To the best of our knowledge, this constitutes the first global existence result for (*) in the biologically most relevant two- and three-dimensional settings. In the proof, we make use of the special structure in (*) and carefully balance the doubly cross-diffusive interaction therein. Indeed, we introduce certain functionals and combine them in a way allowing for cancellations of the most worrisome terms.

AB - We study the system () (equation presented) (inter alia) for D1,D2, χ 1, χ 2, λ 1, λ 2, μ 1, μ 2, a1, a2 > 0 in smooth, bounded domains Ω ⊂ Rn, n ∈{ 1, 2, 3} . Without any further restrictions on these parameters, we prove that there exists a constant stable steady state (u , v ) ∈ [0,∞ )2, meaning that there is ϵ > 0 such that if u0, v0 ∈ W2,2(Ω) are nonnegative with ∂ ν u0 = ∂ ν v0 = 0 in the sense of traces and | u0 u | W2,2(Ω)+| v0 v | W2,2(Ω) < ϵ , then there exists a global classical solution (u, v) of (*) with initial data u0, v0 converging to (u , v ) in W2,2(Ω). Moreover, the convergence rate is exponential, except for the case λ 2μ 1 = λ 1a2, where it is is only algebraical. To the best of our knowledge, this constitutes the first global existence result for (*) in the biologically most relevant two- and three-dimensional settings. In the proof, we make use of the special structure in (*) and carefully balance the doubly cross-diffusive interaction therein. Indeed, we introduce certain functionals and combine them in a way allowing for cancellations of the most worrisome terms.

KW - Double cross diffusion

KW - Large-time behavior

KW - Predator

KW - Prey

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85098579230&partnerID=8YFLogxK

U2 - 10.1137/20M1344536

DO - 10.1137/20M1344536

M3 - Article

VL - 52

SP - 5865

EP - 5891

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 6

ER -