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The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$partial_tu=Delta u-chi abla(u abla v)+u(a(x,t)-ub(x,t)) , 0=Delta v-lambda v+mu u (1)$$where $chi, lambda, mu$ 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 $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0in R$ and every $u_0in C^{b}_{rm unif}(R^N)$, $u_0geq 0$. Next we show the pointwise persistence phenomena in the sense that, for any solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ of (1) with strictly positive initial function $u_0$, then$$0<inf_{t_0in R, tgeq 0}u(x,t+t_0;t_0,u_0)lesup_{t_0in R, tgeq 0} u(x,t+t_0;t_0,u_0)<infty$$and show the uniform persistence phenomena in the sense that there are $0<m<M$ such that for any strictly positive initial function $u_0$, there is $T(u_0)>0$ such that$$mle u(x,t+t_0;t_0,u_0)le M forall,tge T(u_0), xin R^N.$$We then discuss the spreading properties of solutions to (1) with compactly supported initial and prove that there are positive constants $0<c_{-}^{*}le c_{+}^{*}<infty$ such that for every $t_0in R$ and every $u_0in C^b_{rm unif}(R^N), u_0ge 0$ with nonempty compact support, we have that$$lim_{ttoinfty}sup_{|x|ge ct}u(x,t+t_0;t_0,u_0)=0, forall c>c_+^*,$$and$$liminf_{ttoinfty}sup_{|x|le ct}u(x,t+t_0;t_0,u_0)>0, forall 0<c<c_-^*.$$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.
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, $$ begin{cases} partial_tu=Delta u-chi ablacdot(u abla v)+u(a(x,t)-b(x,t)u),quad xin
This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$$partial_tu=Delta u-chi ablacdot(u abla v)+u(a(x,t)-ub(x,t)),quad 0=Delta v-lamb
In this paper, we study traveling wave solutions of the chemotaxis systems begin{equation} begin{cases} u_{t}=Delta u -chi_1 abla( u abla v_1)+chi_2 abla(u abla v_2 )+ u(a -b u), qquad xinmathbb{R} taupartial_tv_1=(Delta- lambda_1 I)v_1+ mu_1 u, q
The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, a
The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauders fixed point theorem.