Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on RN. I. Persistence and asymptotic spreading
Abstract
The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model ∂tu= u-∇(u∇ v)+u(a(x,t)-ub(x,t))\ ,\ 0= v-λ v+μ u \ \ (1)where , 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 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, then0<∈ft0∈ R, t≥ 0u(x,t+t0;t0,u0)t0∈ R, t≥ 0 u(x,t+t0;t0,u0)<∞and show the uniform persistence phenomena in the sense that there are 0<m<M such that for any strictly positive initial function u0, there is T(u0)>0 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 0<c-* c+*<∞ such that for every t0∈ R and every u0∈ Cb unif(RN), u0 0 with nonempty compact support, we have thatt∞|x| ctu(x,t+t0;t0,u0)=0,\ ∀ c>c+*,andt∞|x| ctu(x,t+t0;t0,u0)>0, \ ∀ 0<c<c-*.We also discuss the spreading properties of solutions to (1) with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions and the existence of transition fronts, respectively.
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