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Register a new userOn 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, 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 $c<2sqrt{a}$.
In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.
We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for systems due to Shafrir & Wolansky. Thus for example, when two species interact, one of which has mass less than $4pi$, then the 2-system stays smooth for all time independent of the total mass of the system, in sharp contrast with the well-known breakdown of one specie with initial mass$> 8pi$.
In this paper, we consider the initial Neumann boundary value problem for a degenerate kinetic model of Keller--Segel type. The system features a signal-dependent decreasing motility function that vanishes asymptotically, i.e., degeneracies may take place as the concentration of signals tends to infinity. In the present work, we are interested in the boundedness of classical solutions when the motility function satisfies certain decay rate assumptions. Roughly speaking, in the two-dimensional setting, we prove that classical solution is globally bounded if the motility function decreases slower than an exponential speed at high signal concentrations. In higher dimensions, boundedness is obtained when the motility decreases at certain algebraical speed. The proof is based on the comparison methods developed in our previous work cite{FJ19a,FJ19b} together with a modified Alikakos--Moser type iteration. Besides, new estimations involving certain weighted energies are also constructed to establish the boundedness.
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