2017/09/18 by Salako, Rachidi B., Shen, Wenxian
#35B35 #35B40 #35K57 #35Q92 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1709.05785
The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model ∂tu=Δu-χ∇(u∇ v)+u(a(x,t)-ub(x,t)) , 0=Δv-λv+μu (1)where χ, λ, μ are positive constants, a(x,t) and b(x,t) are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution (u(x,t;t0,u0),v(x,t;t0,u0)) with u(x,t0;t0,u0)=u0(x) for every t0∈ R and every u0∈ Cb\rm unif(RN), u0≥ 0. Next we show the pointwise persistence phenomena in the sense that, for any solution (u(x,t;t0,u0),v(x,t;t0,u0)) of (1) with strictly positive initial function u0, then00 such thatm≤ u(x,t+t0;t0,u0)≤ M ∀ t≥ T(u0), x∈ RN.We then discuss the spreading properties of solutions to (1) with compactly supported initial and prove that there are positive constants 0c+^*,and\liminft→∞sup|x|≤ ctu(x,t+t0;t0,u0)>0, ∀ 0