2018/11/05 by Rachidi B. Salako, Wenxian Shen, Salako, R. B. +1
Mathematics · Medicine · Biochemistry, Genetics and Molecular Biology · #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Evolution and Genetic Dynamics
paper · pdf · doi:10.48550/arxiv.1811.01525
The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the space-time dependent logistic source chemotaxis system, \begincases ∂tu=Δu-χ∇⋅(u∇ v)+u(a(x,t)-b(x,t)u), x∈ RN,\cr 0=Δv-λv+μu , x∈ RN, \endcases (0.1) where N≥ 1 is a positive integer, χ, λ and μ are positive constants, the functions a(x,t) and b(x,t) are positive and bounded. In the first of the series, we studied the phenomena of persistence, and the asymptotic spreading for solutions. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting (0,0) and (u^*(t),v^*(t)) in the case of space homogeneous logistic source. We show that for every χ>0 with χμ(1+\fracsupt∈ Ra(t)inft∈ Ra(t))c*χ and every unit vector ξ, (0.1) has a transition front solution of the form (u(x,t),v(x,t))=(U(x⋅ξ-C(t),t),V(x⋅ξ-C(t),t)) satisfying that C'(t)=\fraca(t)+κ2κ for some number κ>0, \liminft-s→∞(C(t)-C(s))/(t-s)=\underlinec, andlimx→-∞supt∈ R|U(x,t)-u^*(t)|=0 and limx→∞supt∈ R|\fracU(x,t)e-κx-1|=0.Furthermore, we prove that there is no transition front solution (u(x,t),v(x,t))=(U(x⋅ξ-C(t),t),V(x⋅ξ-C(t),t)) of (0.1) connecting (0,0) and (u^*(t),v^*(t)) with least mean speed less than 2√\underlinea, where \underlinea=\liminft-s→∞(1)/(t-s)∫sta(τ)dτ.