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 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.
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.
Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.
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.
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.
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…
Global weak solutions to a chemotaxis model with local sensing and consumption are shown to converge to spatially homogeneous steady states in the large time limit, when the motility is assumed to be positive and $C^1$-smooth on $[0,\infty)$. The result is valid in arbitrary space dimension $n\ge 1$ and extends a previous result which only deals with space dimensions $n\in \{1,2,3\}$.
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.
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 $χα-ξγ>0$.
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.
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…
We show the existence of locally bounded global solutions to the chemotaxis system \[ u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v) \] \[ v_t = Δv - uv \] with homogeneous Neumann boundary conditions and suitably regular positive initial data in smooth bounded domains $Ω\subset \mathbb{R}^N$, $N\geq2$, for $D(u)\geq δu^{m-1}$ with some $δ>0$, provided that $m>1+\frac N4$.
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…
This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant \begin{eqnarray*} \left\{\begin{array}{l} u_t=Δu-\nabla\cdot(u\nabla v), v_t=Δv-uv, \end{array}\right. \end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $Ω\subset\mathbb{R}^n~(n\geq4)$ with smooth boundary. It is proved that if initial data satisfy $u_0\in C^0(\overlineΩ)$ and $v_0\in W^{1,q}(Ω)$ for some $q>n$, the model possesses at least one global renormalized solution.
This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=Δu-χ\nabla\cdot (u\nabla v)+κu-μu^2,\\ v_t=Δv-uv, \end{array} \end{eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $μ$ and prove that for any $μ>0$ there exists a weak solution. Moreover, in the case of $κ>0$ convergence to the constant equilibrium $(\fracκμ,0)$ is shown.
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…
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.
Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, in this article we study solutions to the stationary chemotaxis system \[ \begin{cases} 0 = Δn - \nabla\cdot(n\nabla c) \\ 0 = Δc - nc \end{cases} \] in bounded domains $Ω\subset\mathbb{R}^N$, $N\ge 1$, under no-flux boundary conditions…
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 $s \geq 0$ $$-2 γ^{\prime}(s) + γ^{\prime \prime}(s)s \leq μ_0< μ$$ $$\frac{[γ^{\…
This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, \begin{equation} \begin{cases} u_t=Δu -χ\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N \cr τv_t=Δv-λv +μu,\quad x\in \mathbb{R}^N, \end{cases}(1) \end{equation} where $χ, μ,λ,a,$ and $b$ are positive numbers, and $τ\ge 0$. Among others, it is proved that if $b>2χμ$ and $τ\geq \frac{1}{2}(1-\fracλ{a})_{+} ,$ then for every $c\ge 2\sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{τ,c}(x\cdotξ-ct),V^{τ,c}(x\cdotξ-ct))$ ($\forall\, ξ\in\mathbb…
In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_t=Δu -χ\nabla \cdot (\frac{u}{v} \nabla v) \\ v_t=Δv-f(u)v \end{cases} \end{equation*} in a smooth and bounded domain $Ω$ of $\mathbb{R}^2$, with $χ>0$ and $f\in C^1(\mathbb{R})$ essentially behaving like $u^β$, $0<β<1$. Precisely for $χ<1$ and any sufficiently regular initial data $u(x,0)\geq 0$ and $v(x,0)>0$ on $\barΩ$, we show the existence of global classical solutions. Moreover, if additionally $m:=\int_Ωu(x,0)$ is sufficiently small, then also their boundedness is achieved.
We establish new convergence results, in strong topologies, for solutions of the parabolic-parabolic Keller--Segel system in the plane, to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.
In this work we investigate the following chemo-attraction with consumption model in bounded domains of \, $\mathbb{R}^N$ ($N=1,2,3$): $$ \partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v $$ where $s\ge 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$. The main novelty in the model is the nonlinear potential consumption term $u^sv$. Through the convergence of solutions of an adequate truncated model, two main results are established; existence of uniform in time weak solutions in $3D$ domains, and uniqueness and regularity in $2D…
