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تسجيل مستخدم جديدOn traveling wave solutions in full parabolic Keller-Segel chemotaxis systems with logistic source
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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-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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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
quad 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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