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 $$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}λ_0^{\frac{1}{{s}+1}} (χ+ξ\|w_0\|_{L^\infty(Ω)})}{(\max_{s\geq1}λ_0^{\frac{1}{{s}+1}}(χ+ξ\|w_0\|_{L^\infty(Ω)})-μ)_{+}}+1) (N+\frac{\max_{s\geq1}λ_0^{\frac{1}{{s}+1}}(χ+ξ\|w_0\|_{L^\infty(Ω)})}{(\max_{s\geq1}λ_0^{\frac{1}{{s}+1}} (χ+ξ\|w_0\|_{L^\infty(Ω)})-μ)_{+}}-1)}{N}}}}},$$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global bounded classical solution when $D(0) > 0$ (the case of non-degenerate diffusion), while if, $D(0)\geq 0$ (the case of possibly degenerate diffusion), the existence of bounded weak solutions for system is shown. This extends some recent results by several authors.
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 consider the Neumann problem for a coupled chemotaxis-haptotaxis model of cancer invasion with/without kinetic source in a 2D bounded and smooth domain. For a large class of cell kinetic sources including zero source and sub-logistic sources, we detect an explicit condition involving the chemotactic strength, the asymptotic "damping" rate, and the initial mass of cells to ensure uniform-in-time boundedness for the corresponding Neumann problem. Our finding significantly improves existing 2D global existence and boundedness in related chemotaxis-/haptotaxis systems.
Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)
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…
In this paper, we study the following the coupled chemotaxis--haptotaxis model with remodeling of non-diffusible attractant $$ \left\{\begin{array}{ll} u_t = Δu-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+μu(1- u-w), \displaystyle{v_t=Δv- v +u},\quad \displaystyle{w_t=- vw+ηw(1-u-w)},\quad \end{array}\right. $$ in a bounded smooth domain $\mathbb{R}^2$ with zero-flux boundary conditions, where $χ$, $ξ$ and $η$ are positive parameters. Under appropriate regularity assumptions on the initial data $(u_0, v_0, w_0)$, by develops some $L^p$-estimate techniques, we prove the global existence an…
This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left\{ \begin{array}{llc} u_t=Δu-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w), &(x,t)\in Ω\times (0,T),\\ v_t=Δv-v+u, &(x,t)\inΩ\times (0,T),\\ w_t=-vw,&(x,t)\in Ω\times (0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann boundary conditions. Here $Ω\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and the parameters $ξ,χ,μ>0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $χ/μ$ is sufficiently small, ($\star$) possesses a global class…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nabla u)-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+μu(1- u-w), x\in Ω, t>0,\\ τv_t=Δv- v +u,\quad x\in Ω, t>0,\\ w_t=- vw,\quad x\in Ω, t>0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, where $τ\in\{0,1\}$ and $χ$, $ξ$ and $μ$ are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result…
This work studies the chemotaxis-haptotaxis system $$\left\{ \begin{array}{ll} u_t= Δu - χ\nabla \cdot (u\nabla v) - ξ\nabla \cdot (u\nabla w) + μu(1-u-w), &\qquad x\in Ω, \, t>0, \\[1mm] v_t=Δv-v+u, &\qquad x\in Ω, \, t>0, \\[1mm] w_t=-vw, &\qquad x\in Ω, \, t>0, \end{array} \right. $$ in a bounded smooth domain $Ω\subset\mathbb{R}^2$ with zero-flux boundary conditions, where the parameters $χ, ξ$ and $μ$ are assumed to be positive. It is shown that under appropriate regularity assumption on the initial data $(u_0, v_0, w_0)$, the corresponding initial-boundary problem possesses a unique clas…
This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $Ω\subset \mathbb{R}^N$ ($N=1,2$): $$\label{0} \left\{\begin{array}{ll} p_t=Δp-\nabla\cdotp p(\displaystyle\frac α{1+c}\nabla c+ρ\nabla w)+λp(1-p),\,& x\in Ω, t>0, c_t=Δc-c-μpc,\, &x\in Ω, t>0,\\ w_t= γp(1-w),\,& x\in Ω, t>0, \end{array}\right. $$ where $α, ρ, λ, μ$ and $γ$ are positive parameters. For any reasonably regular initial data $(p_0, c_0, w_0)$, we prove the global boundedness ($L^\infty$-norm) of $p$ via an iterative method. Furthermore, we investigate the long-t…
In this paper, we study the following chemotaxis--haptotaxis system with (generalized) logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+u(a-μu^{r-1}-w), \displaystyle{v_t=Δv- v +u},\quad \\ \displaystyle{w_t=- vw},\quad\\ \displaystyle{\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=\frac{\partial w}{\partial ν}=0},\quad x\in \partialΩ, t>0,\\ \displaystyle{u(x,0)=u_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad x\in Ω, \end{array}\right.\eqno(0.1) $$ %under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathbb{R}^N(…
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.
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 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…
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 $m\in\mathbb{R}$…
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…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big),\\[] 0=Δv+αu-βv,\\[] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$ with smooth boundary $\partial Ω$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. In the case that $m=1$ and $p=q=2$, when $χα-ξγ<0$ and $β=δ$, Tao-Wang (Math. Models Methods Appl. Sci.; 2013; 23; 1-36) proved that global bounded classical solutions toward th…
Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass larg…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partialΩ$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0…
We study an one{dimensional quasilinear system proposed by J. Tello and M. Winkler [19] which models the population dynamics of two competing species attracted by the same chemical. The kinetics terms of the interacting species are chosen to be the Lotka{Volterra type. We prove the existence of global bounded and classical solutions for all chemoattraction rates. Under homogeneous Neumann boundary conditions, we establish the existence of nonconstant steady states by local bifurcation theory. The stability of the bifurcating solutions is also obtained when the diffusivity of both species is la…
We consider a parabolic-elliptic chemotaxis system generalizing \[ \begin{cases}\begin{split} & u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u(u+1)^{σ-1}\nabla v)\\ & 0 = Δv - v + u \end{split}\end{cases} \] in bounded smooth domains $Ω\subset \mathbb{R}^N$, $N\ge 3$, and with homogeneous Neumann boundary conditions. We show that *) solutions are global and bounded if $σ<m-\frac{N-2}N$ *) solutions are global if $σ\le 0$ *) close to given radially symmetric functions there are many initial data producing unbounded solutions if $σ>m-\frac{N-2}N$. In particular, if $σ\le 0$ and $σ> m-\frac{…
Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ τv_t = Δv - v + u \end{cases} \end{align*} in a ball $Ω\subset \mathbb R^n$, $n \ge 2$, wherein $m, q \in \mathbb R$ and $τ\in \{0, 1\}$ are given parameters with $m - q > -1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $L^p(Ω)$ for some $p > p_0 := \frac n2 (1 - (m - q))$. For radially symmetric solutions, we show that, if $u$ is only bounded in $L^{p_0}(Ω)$ and the technical condition $m > \frac{n-2 p_0}{n}$ is fulfil…
This paper is a continuation of the paper Berestycki, Wei and Winter \cite{Berestycki2014}. In \cite{Berestycki2014}, the existence of multiple symmetric and asymmetric spike solutions of a chemotaxis system modeling crime pattern formation, suggested by Short, Bertozzi, and Brantingham \cite{Short2010}, has been proved in the one-dimensional case. The problem of stability of these spike solutions has been left open. In this paper, we establish the existence of a single radial symmetric spike solution for the system in the one and two-dimensional cases. The main difficulty is to deal with quas…
This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $Ω\subseteq \mathbb{R}^3$ with smooth bound…
This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (G(u)χ(v)\nabla v) +\nabla\cdot(H(u)ξ(w)\nabla w), \quad v_t=d_1Δv+αu-βv, \quad w_t=d_2Δw+γu-δw, \quad x \in Ω,\ t>0, \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, α, β, γ, δ>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \…
These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers "Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production" by X. Tao, S. Zhou and M. Ding [J. Math. Anal. Appl. 474:1 (2019) 733-747] and "Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation" by S. Frassu and G. Viglialoro [Acta Appl. Math. 171:1 (2021), 19]. These works, independently published in these l…
