اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTraveling waves of a full parabolic attraction-repulsion chemotaxis systems with logistic sources
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R. B. Salako
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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 $c<2sqrt{a}$.
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-lambda v +mu u,quad xin mathbb{R}^N, end{cases}(1) end{equation} where $chi, mu,lambda,a,$ and $b$ are positive numbers, and $tauge 0$. Among others, it is proved that if $b>2chimu$ and $tau geq frac{1}{2}(1-frac{lambda}{a})_{+} ,$ then for every $cge 2sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{tau,c}(xcdotxi-ct),V^{tau,c}(xcdotxi-ct))$ ($forall, xiinmathbb{R}^N$) connecting the two constant steady states $(0,0)$ and $(frac{a}{b},frac{mu}{lambda}frac{a}{b})$, and there is no such solutions with speed $c$ less than $2sqrt{a}$, which improves considerably the results established in cite{SaSh3}, and shows that (1) has a minimal wave speed $c_0^*=2sqrt a$, which is independent of the chemotaxis.
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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.$$Furthermore, we prove that there is no transition front solution $(u(x,t),v(x,t))=(U(xcdotxi-C(t),t),V(xcdotxi-C(t),t))$ of (0.1) connecting $(0,0)$ and $(u^*(t),v^*(t))$ with least mean speed less than $2sqrt{underline{a}}$, where $underline{a}=liminf_{t-stoinfty}frac{1}{t-s}int_{s}^{t}a(tau)dtau$.
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da v+mu u , xinmathbb{R}^N, (1) $$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 chemotaxis--Navier--Stokes system begin{equation*}label{0.1} left{begin{array}{ll} n_t+ucdot abla n=triangle n-chi ablacdotp left(displaystylefrac n {c} abla cright)+n(r-mu n), c_t+ucdot abla c=triangle c-nc, u_t+ (ucdot abla) u=Delta
u+ abla P+n ablaphi, ablacdot u=0, end{array}right. end{equation*} is considered in a bounded smooth domain $Omega subset mathbb{R}^2$, where $phiin W^{1,infty}(Omega)$, $chi>0$, $rin mathbb{R}$ and $mu> 0$ are given parameters. It is shown that there exists a value $mu_*(Omega,chi, r)geq 0$ such that whenever $ mu>mu_*(Omega,chi, r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $xin Omega$. Moreover, for the case $r>0$, $(n,c,frac {| abla c|}c,u)$ converges to $(frac r mu,0,0,0)$ in $L^infty(Omega)times L^infty(Omega)times L^p(Omega)times L^infty(Omega)$ for any $p>1$ exponentially as $trightarrow infty$, while in the case $r=0$, $(n,c,frac {| abla c|}c,u)$ converges to $(0,0,0,0)$ in $(L^infty(Omega))^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.
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