This paper is concerned with traveling wave solutions for a chemotaxis model with degenerate diffusion of porous medium type. We establish the existence of semi-finite traveling waves, including the sharp type and $C^1$ type semi-finite waves. Our results indicate that chemotaxis slows down the wave speed of semi-finite traveling wave, that is, the traveling wave speed for chemotaxis with porous medium (degenerate) diffusion is smaller than that for the porous medium equation without chemotaxis. As we know, this is a new result not shown in the existing literature. The result appears to be a little surprising since chemotaxis is a connective force. We prove our results by the Schauder's fixed point theorem and estimate the wave speed by a variational approach.
In this note we consider degenerate chemotaxis systems with porous media type diffusion and a source term satisfying the Hadamard growth condition. We prove the Hölder regularity for bounded solutions to parabolic-parabolic as well as for elliptic-parabolic chemotaxis systems.
In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function.
We investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria, in particular, the effect of the initial attractant distribution on bacterial clustering. We prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure, and show the finite speed propagating property without assuming the special structure on attractant concentration, and obtain an explicit formula of the population spreading speed in terms of model parameters. The presented results sugges…
This paper is devoted to the analysis of non-negative solutions for a degenerate parabolic-elliptic Patlak-Keller-Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, there by completing previous results on nite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.
We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of Patlak-Keller-Segel type. We prove its local in time wellposedness in some appropriate space, a blow-up alternative, regularity results and give an idea of the shape of solutions. A transformed and an approximate problem naturally appear in the way of the proof and are also crucial in [22] in order to study the global behaviour of solutions of the equation for a critical pa…
This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =Δn^m - χ\nabla\cdot(n\nabla c) + κn -μn^2$, $x \in Ω,\ t>0$, $c_t + u\cdot\nabla c = Δc - nc$, $x \in Ω,\ t>0$, $u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ, \quad \nabla\cdot u = 0$, $x \in Ω,\ t>0$, where $Ω\subset \mathbb{R}^3$ is a bounded domain and $χ,κ\ge 0$ and $m, μ>0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1…
Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
Osaka Metropolitan University（大阪公立大学）；Institut de Mathématiques de Toulouse (IMT) UMR5219, Université Toulouse III - Paul Sabatier（图卢兹第三大学-保罗·萨巴蒂尔大学）；Université Savoie Mont Blanc, CNRS（萨瓦-蒙布朗大学）
This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{align*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left( \dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), &x\in Ω,\ t>0, \\[1mm] 0 = Δv - μ+ u, &x\in Ω,\ t>0, \end{cases} \end{align*} where $Ω:= B_R(0) \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) is a ball with some $R>0$, and $χ>0$, $p,q\geq1$, $μ:= \frac 1{|Ω|} \int_Ωu_0$ and $u_0$ is an initial data of an unknown function $u$. Bellomo--Winkler (Trans.\ Amer.\ Math.\ Soc.\ Ser.\ B;20…
This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = Δv - μ+ u \end{cases}\end{equation*} under no-flux boundary conditions in balls $Ω\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overlineΩ)$, where $χ>0$ and $μ:=\frac{1}{|Ω|}\int_Ωu_0$. Bellomo--Winkler (Comm.\ Partial Differential Equa…
A class of chemotaxis-Stokes systems generalizing the prototype \[\left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \nabla \cdot \big(n^{m-1}\nabla n\big) - \nabla \cdot \big(n\nabla c\big), c_t + u\cdot\nabla c &=& Δc-nc, u_t +\nabla P &=& Δu + n \nabla ϕ, \qquad \nabla\cdot u =0, \end{array} \right. \] is considered in bounded convex three-dimensional domains, where $ϕ\in W^{2,\infty}(Ω)$ is given. The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion \begin{align*} u_t=&\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w),\\ v_t=&Δv-v+u,\\ w_t=&-vw\end{align*} under homogeneous Neumann boundary conditions in a bounded smooth domain $Ω\subset\mathbb{R}^n$, $n=2, 3, 4$, where $χ, ξ$ and $μ$ are given nonnegative parameters. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geqδu^{m-1}$ for all $u>0$ with some $δ>0$. It is proved that for sufficiently regular initial data global bounde…
This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a $p$-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Hölder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what ext…
This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by \begin{equation} \left\{ \begin{array}{l} \displaystyle u_t=\nabla \cdot \Big(\frac{u\nabla u}{\sqrt{u^2+|\nabla u|^2}}\Big) - χ\, \nabla \cdot \Big(\frac{u\nabla v}{\sqrt{1+|\nabla v|^2}}\Big), \\[1mm] 0=Δv - μ+ u, \end{array} \right. \qquad \qquad (\star) \end{equation} under the initial condition $u|_{t=0}=u_0>0$ and no-flux boundary conditions in a ball $Ω\subset R^n$, where $χ>0$ and $μ:=\frac{1}{|Ω|} \int_Ωu_0$. A previous result [3] has…
This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller-Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic-elliptic system (see the text). Under the initial condition $u|_{t=0}=u_0>0$ and no-flux boundary conditions in balls $Ω\subset\mathbb{R}^n$, where $χ>0$ and $μ:=\frac{1}{|Ω|} \int_Ωu_0$.\abs The main results assert the existence of a uniqu…
A semilinear version of parabolic-elliptic Keller-Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large init…
This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w),\\ v_t=Δv- v +u,\quad \\ w_t=- vw\\ \end{array}\right. $$ in $N$-dimensional smoothly bounded domains, where the parameters $ξ,χ> 0$, $μ> 0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u > 0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it is proved that under the conditions $$m>\frac{2N}{N+{{{\fr…
Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions $d\geq 2$ and in all of space for $d\geq 3$, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of p…
We are concerned with a class of degenerate diffusion equations with time delay describing population dynamics with age structure. In our recent study [{\em Nonlinearity}, 33 (2020), 4013--4029], we established the existence and uniqueness of critical traveling wave for the time-delayed degenerate diffusion equations, and obtained the reducing mechanism of time delay on critical wave speed. In this paper, we now are able to show the asymptotic spreading speed and its coincidence with the critical wave speed $c^*(m,r)$ of sharp wave, and prove that the initial perturbation or the boundary of th…
For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these pro…
