This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \begin{equation*}
\begin{cases}
u_t=\nabla \cdot [(u+α)^{m_1-1}
\nabla u-χu(u+α)^{m_2-2}
\nabla v]
& {\rm in} \; Ω\times (0,T), \\[1mm]
v_t=Δv-v+u
& {\rm in} \; Ω\times (0,T)
\end{cases} \end{equation*} under Neumann boundary conditions and initial conditions, where $Ω$ is a general bounded domain in $\mathbb{R}^n$ with smooth boundary, $α>0$, $χ>0$, $m_1, m_2 \in \mathbb{R}$ and $T>0$. Recently, Anderson-Deng (2017) gave a lower bound for the blow-up time in the case that $m_1=1$ and $Ω$ is a convex bounded domain. The purpose of this paper is to generalize the result in Anderson-Deng (2017) to the case that $m_1 \neq 1$ and $Ω$ is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo-Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of $Ω$.
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the blowup of radially symmetric solutions in terms of suitable Morrey spaces norms are derived.
In this paper we study a two-dimensional chemotaxis-consumption system with singular sensitivity and endowed with Neumann boundary conditions. Sufficient conditions on the data of the problem are given so that the globability of classical solutions is shown, thus excluding any finite-time blow-up scenario.
We study a hyperbolic-parabolic model of chemotaxis in dimensions one and two. In particular, we prove the global existence of classical solutions in certain dissipation regimes.
It is well-known that the Neumann initial-boundary value problem for the minimal-chemotaxis-logistic system in a 2D bounded smooth domain has no blow-up for any choice of parameters. Here, for a large class of kinetic terms including sub-logistic sources, we show that the corresponding 2D Neumann initial-boundary value problems do not possess any blow-up. This illustrates a new phenomenon that even a class of sub-logistic sources can prevent blow-up for the 2D problem, indicating that logistic damping is not the weakest damping to guarantee uniform-in-time boundedness for the 2D minimal Keller…
We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic-parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes.
In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller-Segel chemotaxis-only model possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model. For negligibility of haptotax…
We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v_1) + ξ\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = Δv_1 - βv_1 + αu \\ \partial_t v_2 = Δv_2 - δv_2 + γu, \end{cases} \end{equation*} posed with homogeneous Neumann boundary conditions in bounded domains $Ω=B_R \subset \mathbb{R}^3$, $R>0$, admits radially symmetric solutions which blow-up in finite time if it is attraction-dominated in the sense that $χα-ξγ>0$.
Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.
This paper gives a first insight into making a mathematical bridge between the parabolic-parabolic signal-dependent chemotaxis system and its parabolic-elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic-parabolic chemotaxis system with strong signal sensitivity $$ (u_λ)_t = Δu_λ- \nabla \cdot (u_λχ(v_λ)\nabla u_λ), \quad λ(v_λ)_t = Δv_λ- v_λ+u_λ\quad \mbox{in} \ Ω\times (0,\infty) $$ to that for the parabolic-elliptic chemotaxis system $$ u_t = Δu -\nabla \cdot (uχ(v)\nabla v), \quad 0= Δv -v +u \quad \mbox{in} \ Ω\times (0,\infty), $$ where…
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 paper, we study a free boundary problem for a class of parabolic-elliptic type chemotaxis model in high dimensional symmetry domain Ω. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain Ωwith free boundary condition. Besides, we get the explicit formula for the free boundary and show the chemotactic collapse for the solution of the system.
This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=Δu-χ\nabla \cdot (u\nabla v)+ξ\nabla\cdot(u \nabla w), \quad v_t=Δv-v+u, \quad w_t=Δw-w+u, \quad x \in Ω,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $χ, ξ>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the …
In this paper, we study the minimal Keller-Segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient, especially, chemotactic aggregation to guarantee boundedness and convergence. More specifically, we establish how precisely strong a logistic source can prevent blow-up, and then we obtain an explicit relationship between logistic damping and other ingredient, especially, chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out. Kno…
This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction-repulsion model, with nonlinear productions, diffusion, sensitivities and logistic term, we detect Lebesgue spaces where given unbounded solutions blow-up also in the corresponding norms of those spaces; subsequently, estimates for the blow-up time are established. Finally, for a simplified version of the model, some blow-up criteria are proved.
This work considers a chemotaxis system for multi-species that includes birth or death rate terms, which implies no mass preservation of the populations. We aim to show the convergence to a $L^{\infty}\ -\ $weak solutions, that is local in time, of the JKO - scheme arising from the Optimal Transport Theory, in the spirit of [35,14]. Currently, $L^{\infty}$ solutions have shown to be important in order to get uniqueness. Since death rate case does not ensure global solutions, for arbitrary initial data, in this framework, it could be interest to analyze the Blowing-up phenomenon of this system.…
In this paper we study a semilinear hyperbolic-parabolic system as a model for some chemotaxis phenomena evolving on networks; we consider transmission conditions at the inner nodes which preserve the fluxes and non- homogeneous boundary conditions having in mind phenomena with inflow of cells and food providing at the network exits. We give some conditions on the boundary data which ensure the existence of stationary solutions and we prove that these ones are asymptotic profiles for a class of global solutions.
This paper deals with a parabolic-elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn-Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory by Colli-Visintin [Comm. Partial Differential Equations 15 (1990), 737-756] for a doubly nonlinear evolution inclusion with some bounded monotone operator and subdifferential operator of a proper lower semicontinuous convex function (cf. Colli-Fukao [J. Math. Anal. Appl. 429 (2015), 1190-1213]). Moreover, Colli-Fukao [J. Differential Equations 260 (2016), 6…
This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = Δu - χ\nabla\cdot(u \nabla v) + ξ\nabla\cdot (u \nabla w) + λu - μu^k, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δv + αu - βv, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δw + γu - δw, \quad &x \in Ω,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions, in a ball $Ω\subset \mathbb{R}^n$ ($n \ge 3$), with constant parameters $λ\in \mathbb{R}$, $k>1$, $μ, χ, ξ, α, β, γ, δ>0$. Blow-up phenomena in the system have been well investigated in the c…
This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller--Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order…
In this paper we study a zero-flux attraction-repulsion che\-mo\-taxis-system. We show that despite any mutual interplay between the repulsive and attractive coefficients from the corresponding chemo-sensitivities, even less any restriction on their own sizes, if the production rate of that chemical signal responsible of the cellular coalescence is sublinear, then any initial data emanate a unique global classical solution, which is as well bounded. Further, in a remark of the manuscript, we also address an open question given in \cite{Viglialoro2019RepulsionAttraction}.
In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = Δu - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= Δv -μ+ u , \quad \int_Ωv =0, \ \ μ:= \frac 1 {|Ω|} \int_Ω u dx, \\[6pt] &u(x,0)= u_0(x), \end{aligned} \end{array} \right. \end{equation*} in $Ω\times (0,\infty)$, with $Ω$ a ball in $\mathbb{R}^N$, $N\geq 3$ under homogeneous Neumann boundary conditions and $f(ξ) = (1+ ξ)^{-α}$, $0<α< \frac{N-2}{2(N-1)}$, which describes gradient-dependent limitation of cross diffusion f…
In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis in $\mathbb{R}$. Under some suitable structural assumption on the pressure function, we first predict the system admits a nonlinear diffusion wave in $\mathbb{R}$ based on the empirical results in the literature. Then we show that the solution of the concerned system will locally and asymptotically converges to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergen…
We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.
Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= Δn - \nabla \cdot (n \nabla c), \\ 0 &= Δc - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions \[ \partial_ν c = (γ-c) g \] for $c$ (with $γ>0$ and $C^{1+β}(\partialΩ) \ni g > 0$ for some $β\in(0,1)$) are established in bounded domains $Ω\subset\mathbb{R}^{N}$, $N\ge 1$. Under a smallness condition on $γ$, moreover, we show convergence to the stationary solution.
This paper studies the non-negative solutions of the Keller-Segel model with a nonlocal nonlinear source in a bounded domain. The competition between the aggregation and the nonlocal reaction term is highlighted: when the growth factor is stronger than the dampening effect, with the help of the nonlocal term, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor is of the same order compared to the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse. Global existence of classical solutions is shown for an appropri…
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…
A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity
我们考虑了抛物型趋化模型\[ u_t=Δu - χ\nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=Δv - v + u\]在一个光滑的，有界的，凸的二维区域和显示的整体存在性和有界性的解$χ\in(0,χ_0)$的一些$χ_0>1$，从而证明了值$χ=1$在这方面是不关键的。我们的主要工具是考虑能量泛函\[ \mathcal{F}_{a,b}(u,v)=\int_Ωu\ln u - a \int_Ωu\ln v + b \int_Ω|\nabla \sqrt{v}|^2 \]的$a>0$，$b\geq 0$，其中使用非零值的$b$似乎是新的在这种情况下。
This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, \begin{equation*} \begin{cases} u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v) + λu - μu^κ, & x\inΩ,\ t>0, \\[1mm] 0=Δv - \overline{M_f}(t) + f(u), & x\inΩ,\ t>0, \end{cases} \end{equation*} where $λ>0$, $μ>0$, $κ>1$ and $\overline{M_f}(t):=\frac{1}{|Ω|}\int_Ω f(u(x,t))\,dx$, and $D$, $S$ and $f$ are functions generalizing the prototypes \begin{align*} D(u)=(u+1)^{m-1},\quad S(u)=u(u+1)^{α-1}\quad\mbox{and}\quad f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$…
This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates global-in-time solution. In view of our theorem one sees that one-dimensional Keller- Segel system is essentially different than its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new func…
The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field ar…
In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,\infty}(1\leq j\leq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one spec…
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…
We consider a Parabolic-Elliptic system of PDE's with a chemotactic term in a $N$-dimensional unit ball describing the behavior of the density of a biological species "$u$" and a chemical stimulus "$v$". The system includes a nonlinear chemotactic coefficient depending of ``$\nabla v$", i.e. the chemotactic term is given in the form $$- div (χu |\nabla v|^{p-2} \nabla v), \qquad \mbox{ for } \ p \in ( \frac{N}{N-1},2), \qquad N >2 $$ for a positive constant $χ$ when $v$ satisfies the poisson equation $$- Δv = u - \frac{1}{|Ω|} \int_Ω u_0dx.$$ We study the radially symmetric solutions under the…
In this paper we study classical solutions to the zero--flux attraction--repulsion chemotaxis--system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} u_{ t}=Δu -χ\nabla \cdot (u\nabla v)+ξ\nabla \cdot (u\nabla w) & \textrm{in }Ω\times (0,t^*), \\ 0=Δv+αu-βv & \textrm{in } Ω\times (0,t^*),\\ 0=Δw+γu-δw & \textrm{in } Ω\times (0,t^*),\\ \end{cases} \end{equation} where $Ω$ is a smooth and bounded domain of $\mathbb{R}^2$, $t^*$ is the blow--up time and $α,β,γ,δ,χ,ξ$ are positive real numbers. From the literature it is known that under a proper interplay between the above p…
This paper deals with unbounded solutions to the following zero--flux chemotaxis system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} % about u u_t=\nabla \cdot [(u+α)^{m_1-1} \nabla u-χu(u+α)^{m_2-2} \nabla v] & (x,t) \in Ω\times (0,T_{max}), \\[1mm] % about v 0=Δv-M+u & (x,t) \in Ω\times (0,T_{max}), \end{cases} \end{equation} where $α>0$, $Ω$ is a smooth and bounded domain of $\mathbb{R}^n$, with $n\geq 1$, $t\in (0, T_{max})$, where $T_{max}$ the blow-up time, and $m_1,m_2$ real numbers. Given a sufficiently smooth initial data $u_0:=u(x,0)\geq 0$ and set $M:=\frac…
We study the chemotaxis model $\partial$ t u = div($\nabla$u -- u$\nabla$w) + $θ$v -- u in (0, $\infty$) x $Ω$, $\partial$ t v = u -- $θ$v in (0, $\infty$) x $Ω$, $\partial$ t w = D$Δ$w -- $α$w + v in (0, $\infty$) x $Ω$, with no-flux boundary conditions in a bounded and smooth domain $Ω$ $\subset$ R 2 , where u and v represent the densities of subpopulations of moving and static individuals of some species, respectively, and w the concentration of a chemoattractant. We prove that, in an appropriate functional setting, all solutions exist globally in time. Moreover, we establish the existence …
