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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,$$partial_tu=Delta u-chi ablacdot(u abla v)+u(a(x,t)-ub(x,t)),quad 0=Delta v-lambda v+mu u , xinmathbb{R}^N, (1) $$where $Nge1$ is a positive integer, $chi,lambda,mu>0$, and the functions $a(x,t), b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In this direction, we prove that, if $0lemuchi<b_inf$, then (1) has a strictly positive entire solution, which is time-periodic (respectively time homogeneous) when the logistic source function is time-periodic (respectively time homogeneous). Next, we show that there is positive constant $chi_0$, such that for every $0lechi<chi_0$, (1) has a unique positive entire solution which is uniform and exponentially stable with respect to strictly positive perturbations. In particular, we prove that $chi_0$ can be taken to be $b_inf/2mu$ when the logistic source function is either space homogeneous or the function $b(x,t)/a(x,t)$ is constant. We also investigate the disturbances to Fisher-KKP dynamics caused by chemotatic effects, and prove that$$sup_{0<chilechi_1}sup_{t_0inmathbb{R},tge 0}frac{1}{chi}|u_{chi}(cdot,t+t_0;t_0,u_0)-u_0(cdot,t+t_0;t_0,u_0)|_{infty}<infty$$for every $0<chi_1<b_inf/mu$ and every uniformly continuous initial function $u_0$, with $u_{0inf}>0$, where $(u_chi(x,t+t_0;t_0,u_0),v_chi(x,t+t_0;t_0,u_0))$ denotes the unique classical solution of (1) with $u_chi(x,t_0;t_0,u_0)=u_0(x)$, for every $0lechi<b_inf$.
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, $$ begin{cases} partial_tu=Delta u-chi ablacdot(u abla v)+u(a(x,t)-b(x,t)u),quad xin
The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$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,
This paper is devoted to discussing the existence and uniqueness of weak solutions to time-fractional elliptic equations having time-dependent variable coefficients. To obtain the main result, our strategy is to combine the Galerkin method, a basic i
This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, begin{equation} begin{cases} u_t=Delta u -chi ablacdot(u abla v)+u(a-bu),quad xinmathbb{R}^N cr tau v_t=Delta v
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