Rafael Granero-Belinchón
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.
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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.
Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system
当初始条件的平均值超过某个门限值时，一维拟线性抛物-抛物趋化系统的解将在有限时间内爆破.证明结合了一个新的身份维里类型的有界性从下面的Liapunov功能与系统，后者是特有的一维设置。
Global entropy solutions to a degenerate parabolic-parabolic chemotaxis system for flux-limited dispersal
本文在有界区域无通量边界条件下，针对通量饱和的退化抛物-抛物趋化系统，利用Schauder不动点定理和有界变差函数方法证明了全局有限时间有界熵解的存在性。
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 work we consider the Keller-Segel model for chemotaxis on networks, both in the doubly parabolic case and in the parabolic-elliptic one. Introducing appropriate transition conditions at vertices, we prove the existence of a time global and spatially continuous solution for each of the two systems. The main tool is the use of the explicit formula for the fundamental solution of the heat equation on a weighted graph and of the corresponding sharp estimates.
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.
We consider a general hyperbolic model of chemotaxis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem and we determine their asymptotic behavior. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts and using detailed decay estimates of the Green function.
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.
We consider a parabolic-parabolic Keller-Segel system of chemotaxis model with singular sensitivity $u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)$, $v_t=kΔv-v+u$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\geq2)$, with $χ,k>0$. It is proved that for any $k>0$, the problem admits global classical solutions, whenever $χ\in\big(0,-\frac{k-1}{2}+\frac{1}{2}\sqrt{(k-1)^2+\frac{8k}{n}}\big)$. The global solutions are moreover globally bounded if $n\le 8$. This shows an exact way the size of the diffusion constant $k$ of the chemicals $v$ effects th…
In two space dimensions, the parabolic-parabolic Keller--Segel system shares many properties with the parabolic-elliptic Keller--Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8?. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8? always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller--Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8?, which…
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…
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…
Global boundedness induced by asymptotically non-degenerate motility in a fully parabolic chemotaxis model with local sensing
本文研究了具有局部感知的Keller-Segel型全抛物线趋化模型，通过引入依赖信号的渐近非退化迁移率函数描述排斥主导的趋化性，证明了任意空间维度下经典解的全局有界性，并在迁移率单调非减时验证了向齐次稳态的稳定化。
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.
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 $χα-ξγ>0$.
Boundary effect on asymptotic behaviour of solution to the hyperbolic-parabolic chemotaxis system
本文研究半直线上拟线性双曲-抛物趋化系统解的全局存在性与稳定性，在两种边界条件下建立渐近状态方程解的存在唯一性，证明原方程解的非线性渐近稳定性，并给出收敛速率。
Global existence and boundedness of solutions to a fully parabolic chemotaxis system with indirect signal production in $\mathbb{R}^4$
本文研究四维全抛物型趋化系统，证明初值质量低于$(8\pi)^2$时解整体存在，并给出保证解一致有界的初值质量条件。
Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source
本文研究二维有界区域中带空间异质逻辑源的全抛物趋化-增长系统，证明了爆破点集合包含在阻尼系数μ的零点集合内，揭示了空间异质性对爆破位置的约束。
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 consider the parabolic chemotaxis model \[ u_t=Δu - χ\nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=Δv - v + u\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $χ\in(0,χ_0)$ for some $χ_0>1$, thereby proving that the value $χ=1$ is not critical in this regard. Our main tool is consideration of the energy functional \[ \mathcal{F}_{a,b}(u,v)=\int_Ωu\ln u - a \int_Ωu\ln v + b \int_Ω|\nabla \sqrt{v}|^2 \] for $a>0$, $b\geq 0$, where using nonzero values of $b$ appears to be new in this context.
Boundedness through nonlocal dampening effects in a fully parabolic chemotaxis model with sub and superquadratic growth
本文研究了包含次和超二次增长外部源及非局部阻尼反应的趋化模型，证明了在有界不可渗透区域中，只要阻尼效应足够强，细胞就不会随时间出现聚集效应。
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…
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 …
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…
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…
Explicit lower bound of blow-up time in a fully parabolic attraction-repulsion chemotaxis system with nonlinear terms
本文针对空间维数大于3的全抛物Keller-Segel型系统，证明高能量水平下能量函数的爆破时间为经典爆破时间，并给出带非线性项的全抛物吸引-排斥趋化系统爆破时间的显式下界估计。
Relationship between haptotaxis and chemotaxis in cell dynamics
本研究探讨了细胞分选中趋触性和趋化性之间的关系，通过分析非局部聚集模型与Keller-Segel型趋化系统之间的联系，揭示了两种机制如何产生相似动态行为。
An Euler-type hyperbolic-parabolic system of chemotactic aggregation describing the vascular network formation is investigated in the critical regularity setting. For small initial data around a constant equilibrium state, the well-posedness of the global classical solution to the Cauchy problem with general pressure laws is established in homogeneous hybrid Besov spaces. Then, the optimal time-decay rates of the global solution are analyzed under an additional regularity assumption on the initial data. Furthermore, the relaxation limit (large friction limit) of the hyperbolic-parabolic system…
Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis
本文研究趋化性抛物-双曲系统在强稀疏波扰动下的渐近行为，通过能量法证明小扰动初值问题存在唯一全局解且随时间收敛到稀疏波，且结果与波强度无关。
Absence of blow-up in a fully parabolic chemotaxis system with weak singular sensitivity and logistic damping in dimension two
本文研究了二维全抛物趋化系统在弱奇异敏感性和逻辑阻尼下的爆破问题，证明了在齐次 Neumann 边界条件下不会发生爆破，并将已知抛物-椭圆情形的结论推广至全抛物情形。
Global existence and stability in a class of chemotaxis systems with lethal interactions, nonlinear diffusion and production
本文研究了具有致死相互作用、非线性扩散和生成的化学趋向系统，证明了在适当参数条件下，该系统在任意空间维度下均存在唯一的全局有界经典解，并通过Lyapunov函数分析了解的渐进行为稳定性。
The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N},\cr {v_t}=Δv -λv+μu,\quad x\in \mathbb{R}^{N}. \end{cases}(1) \end{equation} where $χ, \ a,\ b,\ λ,\ μ$ are positive constants. Assume $b>\frac{Nμχ}{4}$. Among others, it is proved that $2\sqrt{a}$ is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is, $$ \lim_{t\to\infty}\sup…
This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=Δu-\nabla \cdot (uχ(v)\nabla v) +\nabla \cdot (uξ(w)\nabla w), &x \in Ω,\ t>0,\\[1.05mm] v_t=Δv-v+u, &x \in Ω,\ t>0,\\[1.05mm] w_t=Δw-w+u, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with smooth boundary, $χ, ξ$ are functions satisfying some conditions. Global existence and boundedness of classical solutions to the syst…
In this article we investigate a parabolic-parabolic-elliptic two-species chemotaxis system with weak competition and show global asymptotic stability of the coexistence steady state under a smallness condition on the chemotactic strengths, which seems more natural than the condition previously known. For the proof we rely on the method of eventual comparison, which thereby is shown to be a useful tool even in the presence of chemotactic terms.
Boundary symmetry breaking via logistic damping in a chemotaxis-growth system
研究在1D领域内具有对数敏感性的趋化-生长模型的全局稳定性，分析抛物线-抛物线和抛物线-双曲线系统，通过引入动态参考轮廓处理时间依赖的边界数据，证明能量估计、解的有界性和扰动的全局渐进行为。
Finite-time blowup in a parabolic-parabolic-elliptic chemotaxis model involving indirect signal production
本文研究了涉及间接信号产生的趋化模型，证明了在特定初始条件下，解在有限时间内爆破。
We consider a parabolic-elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$ \left\{ \begin{array}{l} u_t -Δ(u γ(v)= μu(1-u), \\ - Δv +v=u, \end{array} \right. $$ under Neumann boundary conditions and appropriate initial data in a bounded and regular domain $Ω$ of $\R^N$ (for $N \geq 1)$, where $γ\in C^3([0, \infty))$ and satisfies the assumptions $γ(s) > 0$, $γ^{\prime}(s) \leq 0$, $γ^{\prime \prime} (s) \geq 0$, $γ^{\prime \prime \prime}(s) \leq 0$ for any $s \geq 0$ $$-2 γ^{\prime}(s) + γ^{\prime \prime}(s)s \leq μ_0< μ$$ $$\frac{[γ^{\…
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=Δu -χ\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N \cr τv_t=Δv-λv +μu,\quad x\in \mathbb{R}^N, \end{cases}(1) \end{equation} where $χ, μ,λ,a,$ and $b$ are positive numbers, and $τ\ge 0$. Among others, it is proved that if $b>2χμ$ and $τ\geq \frac{1}{2}(1-\fracλ{a})_{+} ,$ then for every $c\ge 2\sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{τ,c}(x\cdotξ-ct),V^{τ,c}(x\cdotξ-ct))$ ($\forall\, ξ\in\mathbb…
On the parabolic-elliptic limit of the doubly parabolic Keller--Segel system modelling chemotaxis
在强拓扑中，当一个物理参数趋于零时，我们建立了抛物-抛物Keller-Segel方程组在平面上的解到抛物-椭圆模型的相应解的新的收敛结果.我们的主要工具是合适的时空估计，这意味着这些模型的缓慢衰减（一般来说，不可积）的解决方案的整体存在，在自然的小假设。
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…
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…
Upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak-Keller-Segel model of chemotaxis
本文在仅假设初始数据非负且质量有限的条件下，给出了二维抛物-椭圆Patlak-Keller-Segel趋化系统爆破临界时间的上界，结果同时依赖于超临界质量和初始数据形状。
The paper should be viewed as complement of an earlier result in [8]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form 1/u all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [8] deals with the Jager-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [8], being a main step of th…
On the existence of solutions for a parabolic-elliptic chemotaxis model with flux limitation and logistic source
本文研究带通量限制趋化项与Logistic源的抛物-椭圆趋化系统，在齐次Neumann边界条件下证明了当$p<3/2$、空间维数$N\ge2$时系统存在一致有界解，丰富了趋化模型解的存在性理论。
Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments
本文研究了带logistic源的抛物型-抛物型趋化模型在异质有界域中的渐进行为，证明了在特定参数条件下系统存在唯一的正整体解，并且该解全局渐近稳定。
The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= Δu-χ\nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N τv_t=Δv-v+u, \quad x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ represents the population density of a mobile species and $v(x,t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We prove that for every $τ>0$, there is $0<χ_τ^*<\frac{b}{2}$ such that for every $0<χ<χ_τ^*$, there exist two positive numbers $2\sqrt a \le c^{*}(χ…
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the existence of radial global-in-time solutions in terms of suitable Morrey norms are derived.
This paper deals with the two-species chemotaxis-competition system $u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u(1 - u - a_1 v)$, $v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v(1 - a_2 u - v)$, $0 = d_3 Δw + αu + βv - γw$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $χ_i$ and $μ_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{χ_i}{μ_i}$ are small. However, the conditions in the above…
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 classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=Δ(γ(v) u ) &\mathrm{in}\ Ω\times(0,\infty), \\[1mm] v_t=Δv - v + u &\mathrm{in}\ Ω\times(0,\infty), \\[1mm] \displaystyle \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 &\mathrm{on}\ \partialΩ\times (0,\infty), \\[1mm] u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 &\mathrm{in}\ Ω, \end{cases} \end{align*} where $Ω$ is a smooth bounded domain in $\mathbf{R}^n$($n \geq 3$), $γ(v)=v^{-k}$ ($k>0$) and the initial data $(u_0,v_0)$ is positive and regular. This system has striking featu…
Global existence of solutions to the fully parabolic chemotaxis system with logistic source under nonlinear Neumann boundary conditions
本文研究了在非线性Neumann边界条件下具有logistic源的全抛物趋化系统，证明了在任意空间维度$n \geq 2$且$p<\frac{3}{2}$时全局有界解的存在性，扩展了Le（2023）仅限于$n=2$和$n=3$的结论。
An improvement toward global boundedness in a fully parabolic chemotaxis with singular sensitivity in any dimension
本文在任意维度中改进了全抛物趋化系统的全局有界性结果，证明在特定χ条件下存在全局有界解，推翻了原有阈值的最优性假设。
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.
A sharp criterion and complete classification of global-in-time solutions and finite time blow-up of solutions to a chemotaxis system in supercritical dimensions
本文研究了超临界维度中化学趋化系统的严格准则和完整分类，通过尺度临界Morrey范数确定解的长期行为，揭示了新的临界指数和奇异稳态解的存在。
Nonlinear stability of composite waves of traveling wave and rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis
本文研究趋化性抛物-双曲系统中行波与稀疏波复合波的非线性稳定性，通过加权相对熵与能量结构结合，证明小扰动下全局解收敛于复合波。
We examine a general model for the Fisher-KPP (FKPP) equation with nonlocal advection. The main interpretation of this model is as describing a diffusing and logistically growing population that is also influenced by intraspecific attraction or repulsion. For a particular choice of parameters, this specializes to the Keller-Segel-Fisher equation for chemotaxis. Our interest is in the effect of chemotaxis on the speed of traveling waves. We prove that there is a threshold such that, when interactions are weaker and more localized than this, chemotaxis, despite being non-trivial, does not influe…
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 $m_1=1$ an…
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