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Register a new userParabolic-elliptic chemotaxis model with space-time dependent logistic sources on $mathbb{R}^N$. I. Persistence and asymptotic spreading
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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.
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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, $$ begin{cases} partial_tu=Delta u-chi ablacdot(u abla v)+u(a(x,t)-b(x,t)u),quad xin R^N,cr 0=Delta v-lambda v+mu u ,quad xin R^N, end{cases} (0.1) $$ where $Nge 1$ is a positive integer, $chi, lambda$ and $mu$ 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 $chi>0$ with $chimubig(1+frac{sup_{tin R}a(t)}{inf_{tin R}a(t)}big)<inf_{tin R}b(t)$, there is a positive constant $c^{*}_{chi}$ such that for every $underline{c}>c^{*}_{chi}$ and every unit vector $xi$, (0.1) has a transition front solution of the form $(u(x,t),v(x,t))=(U(xcdotxi-C(t),t),V(xcdotxi-C(t),t))$ satisfying that $C(t)=frac{a(t)+kappa^2}{kappa}$ for some number $kappa>0$, $liminf_{t-stoinfty}frac{C(t)-C(s)}{t-s}=underline{c}$, and$$lim_{xto-infty}sup_{tin R}|U(x,t)-u^*(t)|=0 quad text{and}quad lim_{xtoinfty}sup_{tin R}|frac{U(x,t)}{e^{-kappa x}}-1|=0.$$Furthermore, we prove that there is no transition front solution $(u(x,t),v(x,t))=(U(xcdotxi-C(t),t),V(xcdotxi-C(t),t))$ of (0.1) connecting $(0,0)$ and $(u^*(t),v^*(t))$ with least mean speed less than $2sqrt{underline{a}}$, where $underline{a}=liminf_{t-stoinfty}frac{1}{t-s}int_{s}^{t}a(tau)dtau$.
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 $0lechi<b_inf$.
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, qquad xinmathbb{R}, taupartial v_2=(Delta- lambda_2 I)v_2+ mu_2 u, qquad xinmathbb{R}, end{cases} (0.1) end{equation} where $tau>0,chi_{i}> 0,lambda_i> 0, mu_i>0$ ($i=1,2$) and $ a>0, b> 0$ are constants, and $N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $ 0<c^{*}(tau,chi_1,mu_1,lambda_1,chi_2,mu_2,lambda_2)<c^{**}(tau,chi_1,mu_1,lambda_1,chi_2,mu_2,lambda_2)$ such that for every $c^{*}(tau,chi_1,mu_1,lambda_1,chi_2,mu_2,lambda_2)leq c<c^{**}(tau,chi_1,mu_1,lambda_1,chi_2,mu_2,lambda_2)$, $(0.1)$ has a traveling wave solution $(u,v_1,v_2)(x,t)=(U,V_1,V_2)(x-ct)$ connecting $(frac{a}{b},frac{amu_1}{blambda_1},frac{amu_2}{blambda_2})$ and $(0,0,0)$ satisfying $$ lim_{zto infty}frac{U(z)}{e^{-mu z}}=1, $$ where $muin (0,sqrt a)$ is such that $c=c_mu:=mu+frac{a}{mu}$. Moreover, $$ lim_{(chi_1,chi_2)to (0^+,0^+))}c^{**}(tau,chi_1,mu_1,lambda_1,chi_2,mu_2,lambda_2)=infty$$ and $$lim_{(chi_1,chi_2)to (0^+,0^+))}c^{*}(tau,chi_1,mu_1,lambda_1,chi_2,mu_2,lambda_2)= c_{tilde{mu}^*}, $$ where $tilde{mu}^*={min{sqrt{a}, sqrt{frac{lambda_1+tau a}{(1-tau)_{+}}},sqrt{frac{lambda_2+tau a}{(1-tau)_{+}}}}}$. We also show that $(0.1)$ has no traveling wave solution connecting $(frac{a}{b},frac{amu_1}{blambda_1},frac{amu_2}{blambda_2})$ and $(0,0,0)$ with speed $c<2sqrt{a}$.
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, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$.
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.
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