We study the singular limit of a system of partial differential equations which is a model for an aggregation of amoebae subjected to three effects: diffusion, growth and chemotaxis. The limit problem involves motion by mean curvature together with a nonlocal drift term. We consider rather general initial data. We prove a generation of interface property and study the motion of interfaces. We also obtain an optimal estimate of the thickness and the location of the transition layer that develops.
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.
Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn-Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn-Hilliard-Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.
In this paper, we study the spatial propagation dynamics of a parabolic-elliptic chemotaxis system with logistic source which reduces to the well-known Fisher-KPP equation without chemotaxis. It is known that for fast decaying initial functions, this system has a finite spreading speed. For slowly decaying initial functions, we show that the accelerating propagation will occur and chemotaxis does not affect the propagation mode determined by slowly decaying initial functions if the logistic damping is strong, that is, the system has the same upper and lower bounds of the accelerating propagati…
Introducing a suitable solution concept, we show that in bounded smooth domains $Ω\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=Δu -χ\nabla\cdot\left(\frac{u}{v}\nabla v\right)+κu -μu^2,\\ v_t&=Δv -uv, \end{align*} with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any $μ, κ, χ>0$.
In this paper, we study the initial-boundary value problem and its asymptotic behavior for a repulsive chemotaxis model with logarithmic sensitivity and logistic growth. We establish global well-posedness of strong solutions for large initial data with Neumann boundary conditions and, moreover, establish the qualitative result that both the population density and chemical concentration asymptotically converge to constant states with the population density specifically converging to its carrying capacity. We additionally prove that the vanishing chemical diffusivity limit holds in this regime. …
Assuming that $0<χ<\sqrt{\frac{2}n}$, $κ\ge 0$ and $μ>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= Δu - χ\nabla\cdot ( \frac{u}{v} \nabla v ) + κu -μu^2\\ v_t &= Δv - u v \end{split} \] in a bounded domain $Ω\subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $χ>0$, $μ>0$, $κ\ge 0$.
A considerable number of research works has been devoted to the study of tumor models. Several biophysical factors, such as cell proliferation, apoptosis, chemotaxis, angiogenesis and necrosis, have been discovered to have an impact on the complicated biological system of tumors. An indicator of the aggressiveness of tumor development is the instability of the shape of the tumor boundary. Complex patterns of tumor morphology have been explored by Lu, Min-Jhe et al. [Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis, Journal of Computational Physics 459 (…
This paper is concerned with the asymptotic stability of the initial-boundary value problem of a singular PDE-ODE hybrid chemotaxis system in the half space $\R_+=[0, \infty)$. We show that when the non-zero flux boundary condition at $x=0$ is prescribed and the initial data are suitably chosen, the solution of the initial-boundary value problem converges, as time tend to infinity, to a shifted traveling wavefront restricted in the half space $[0,\infty)$ where the wave profile and speed are uniquely selected by the boundary flux data. The results are proved by a Cole-Hopf type transformation …
Mathematical models that describe the tumor growth process have been formulated by several authors in order to understand how cancer develops and to develop new treatment approaches. In this study, it is aimed to investigate the long-time behavior of the two-phase diffuse-interface model, which was proposed in [11] to model a tumor tissue as a mixture of cancerous and healthy cells. Up to now, studies on the long-time behavior of this model have neglected chemotaxis and active transport, which have a significant effect on tumor growth. In this research, our main aim is to study this model with…
A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differenti…
We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effect. The PDE system couples a Navier-Stokes equation for the fluid velocity, a convective Cahn-Hilliard equation for the phase field variable with an advection-diffusion-reaction equation for the nutrient density. For the system with a singular potential, we prove the existence of global weak solutions in both two and three dimensions. Besides, in the two dimensional case, we establish a continuous dependence result that implies the uniqueness of global weak solutions. The sing…
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.
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 is concerned with the well-posedness and large-time behavior of a two dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial datum…
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…
We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when …
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^{\…
In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)+ru-μu^k$, $0=Δv-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $Ω\subset\mathbb{R}^n$, $χ,r,μ>0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{χ^2}{4}$ for $0<χ\le 2$, or $r> χ-1$ for $χ>2$. In addition, under the same condition for $r,χ$, the system admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3…
This paper is concerned with the following singularly perturbed non-local semi-linear problem \begin{equation} \label{h} \tag{$\ast$} \begin{cases} \varepsilon^2 Δu=\frac{m}{\int_Ωe^{u}{\mathrm{d}x}}u e^u\quad &\mathrm{in}~Ω,\\ u= u_0~&\mathrm{on}~\partialΩ, \end{cases} \end{equation} which corresponds to the stationary problem of a chemotaxis system describing the aerobic bacterial movement, where $Ω$ is a smooth bounded domain in $\mathbb{R}^N (N\geq 1)$, $\varepsilon, m$ and $u_0$ are positive constants. We show that the problem \eqref{h} admits a unique classical solution which is of bound…
New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose.
This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by long-range interactions. These integro-partial differential equations model cell-to-cell adhesion by a non-local term and may be seen as non-local variants of the corresponding local model proposed by H. Garcke et al. (2016). The model in consideration couples a non-local Cahn-Hilliard equation for the tumor phase variable with a reaction-diffusion equation for the nutrient concentration, and takes into account also significant mechanisms such a…
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…
We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associat…
