A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant

In this paper, we study the following the coupled chemotaxis--haptotaxis model with remodeling of non-diffusible attractant $$\left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+\mu u(1- u-w), \displaystyle{v_t=\Delta v- v +u},\quad \displaystyle{w_t=- vw+\eta w(1-u-w)},\quad \end{array}\right.$$ in a bounded smooth domain $\mathbb{R}^2$ with zero-flux boundary conditions, where $\chi$, $\xi$ and $\eta$ 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 and uniqueness of classical solutions when $\mu>0$ (where $\mu$ is the logistic growth rate of cancer cells). Here we use an approach based on maximal Sobolev regularity and the variation-of-constants formula remove the restrictions $\mu$ is sufficiently large, which required in \cite{PangPang1}

Boundedness of solution of a parabolic--ODE--parabolic chemotaxis--haptotaxis model with (generalized) logistic source

In this paper, we study the following chemotaxis--haptotaxis system with (generalized) logistic source \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), \displaystyle{v_t=\Delta v- v +u},\quad \\ \displaystyle{w_t=- vw},\quad\\ \displaystyle{\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial w}{\partial \nu}=0},\quad x\in \partial\Omega, t>0,\\ \displaystyle{u(x,0)=u_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad x\in \Omega, \end{array}\right.\eqno(0.1) %under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathbb{R}^N(N\geq1)$, with parameter $r>1$. the parameters $a\in \mathbb{R}, \mu>0, \chi>0$. It is shown that when $r>2$, or \begin{equation*} \mu>\mu^{*}=\begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1},~~~\mbox{if}~~r=2, \end{array} \end{equation*} % $\mu>\frac{(N-2)_{+}}{N}\chi C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, the considered problem possesses a global classical solution which is bounded, where $C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$ is a positive constant which is corresponding to the maximal sobolev regularity. Here $C_{\beta}$ is a positive constant which depends on $\xi$, $\|u_0\|_{C(\bar{\Omega})},\|v_0\|_{W^{1,\infty}(\Omega)}$ and $\|w_0\|_{L^\infty(\Omega)}$. This result improves or extends previous results of several authors.Comment: arXiv admin note: text overlap with arXiv:1711.1004

A new (and optimal) result for boundedness of solution of a quasilinear chemotaxis--haptotaxis model (with logistic source)

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+\mu u(1- u-w), x\in \Omega, t>0,\\ \tau v_t=\Delta v- v +u,\quad x\in \Omega, t>0,\\ w_t=- vw,\quad x\in \Omega, t>0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, where $\tau\in\{0,1\}$ and $\chi$, $\xi$ and $\mu$ are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result which (precisely) gives the relationship between $m,\xi,\chi$ and $\mu$ that yields to the boundedness of the solutions. Moreover, these results thereby significantly extending results of previous results of several authors (see Remarks 1.1 and 1.2) and some optimal results are obtained.Comment: 4

Boundedness in a two-dimensional chemotaxis-haptotaxis system

This work studies the chemotaxis-haptotaxis system $$\left\{ \begin{array}{ll} u_t= \Delta u - \chi \nabla \cdot (u\nabla v) - \xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), &\qquad x\in \Omega, \, t>0, \\[1mm] v_t=\Delta v-v+u, &\qquad x\in \Omega, \, t>0, \\[1mm] w_t=-vw, &\qquad x\in \Omega, \, t>0, \end{array} \right.$$ in a bounded smooth domain $\Omega\subset\mathbb{R}^2$ with zero-flux boundary conditions, where the parameters $\chi, \xi$ and $\mu$ 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 classical solution which is global in time and bounded. In addition to coupled estimate techniques, a novel ingredient in the proof is to establish a one-sided pointwise estimate, which connects $\Delta w$ to $v$ and thereby enables us to derive useful energy-type inequalities that bypass $w$. However, we note that the approach developed in this paper seems to be confined to the two-dimensional setting.Comment: 17 page

Boundedness in a three-dimensional chemotaxis-haptotaxis model

This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left\{ \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ w_t=-vw,&(x,t)\in \Omega\times (0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann boundary conditions. Here $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and the parameters $\xi,\chi,\mu>0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $\chi/\mu$ is sufficiently small, ($\star$) possesses a global classical solution which is bounded in $\Omega\times(0,\infty)$. We underline that the result fully parallels the corresponding parabolic-elliptic-ODE system.Comment: correct Lemma 2.5 in version 1 due to an error in the proof, and reform Sec.3 to be more clear, main results and arguments remain unchange

Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion

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)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w),\\ v_t=&\Delta v-v+u,\\ w_t=&-vw\end{align*} under homogeneous Neumann boundary conditions in a bounded smooth domain $\Omega\subset\mathbb{R}^n$, $n=2, 3, 4$, where $\chi, \xi$ and $\mu$ are given nonnegative parameters. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq\delta u^{m-1}$ for all $u>0$ with some $\delta>0$. It is proved that for sufficiently regular initial data global bounded solutions exist whenever $m>2-\frac{2}{n}$. For the case of non-degenerate diffusion (i.e. $D(0)>0$) the solutions are classical; for the case of possibly degenerate diffusion ($D(0)\geq 0$), the existence of bounded weak solutions is shown

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)

We propose and study a class of parabolic-ODE models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumor microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature consituting a core difference to both classical Keller-Segel chemotaxis systems and Chaplain-Lolas type chemotaxis-haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.Comment: 36pages, 2 figure

Singular structure formation in a degenerate haptotaxis model involving myopic diffusion

We consider the system $$u_t=\big(d(x)u\big)_{xx} - \big(d(x)uw_x\big)_x, \quad w_t=-ug(w),$$ which arises as a simple model for haptotactic migration in heterogeneous environments, such as typically occurring in the invasive dynamics of glioma. A particular focus is on situations when the diffusion herein is degenerate in the sense that the zero set of $d$ is not empty. It is shown that if such possibly present degeneracies are sufficiently mild in the sense that $$\int_\Omega \frac{1}{d}<\infty,$$ then under appropriate assumptions on the initial data a corresponding initial-boundary value problem, posed under no-flux boundary conditions in a bounded open real interval $\Omega$, possesses at least one globally defined generalized solution. Moreover, despite such degeneracies the considered myopic diffusion mechanism is seen to asymptotically determine the solution behavior in the sense that for some constant $\mu_\infty>0$, the obtained solution satisfies u(\cdot,t)\rightharpoonup \frac{\mu_\infty}{d} \ \mbox{in } L^1(\Omega) \quad \mbox{and} \quad w(\cdot,t) \to 0 \ \mbox{in } L^\infty(\Omega) \quad \mbox{as } t\to\infty, \qquad (\star) and that hence in the degenerate case the solution component $u$ stabilizes toward a state involving infinite densities, which is in good accordance with experimentally observed phenomena of cell aggregation. Finally, under slightly stronger hypotheses inter alia requiring that $\frac{1}{d}$ belong to $L\log L(\Omega)$, a substantial effect of diffusion is shown to appear already immediately by proving that for a.e.~$t>0$, the quantity $\ln (du(\cdot,t))$ is bounded in $\Omega$. In degenerate situations, this particularly implies that the blow-up phenomena expressed in ($\star$) in fact occur instantaneously

Negligibility of haptotaxis effect in a chemotaxis-haptotaxis model

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 haptotaxis on asymptotical behavior, we show that any global-in-time haptotaxis solution component vanishes exponentially as time approaches infinity, and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, the aforementioned critical mass blow-up phenomenon for the chemotaxis-only model is almost undestroyed even with arbitrary introduction of haptotaixs, showing negligibility of haptotaxis effect compared to chemotaxis effect in terms of boundedness, blow-up and longtime behavior in the chemotaxis-haptotaxis model.Comment: 30 pages;under review in a journa

Global solvability and boundedness in the NN-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in \Omega, t>0, \disp{v_t-\Delta v=-uv },\quad x\in \Omega, t>0, %\disp{\tau w_t+\delta w=u },\quad %x\in \Omega, t>0, \disp{(\nabla D(u)-\chi u\cdot \nabla v)\cdot \nu=\frac{\partial v}{\partial\nu}=0},\quad x\in \partial\Omega, t>0, \disp{u(x,0)=u_0(x)},\quad v(x,0)=v_0(x),~~ x\in \Omega \end{array}\right. on a bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partial\Omega, \chi$ and $\mu$ are positive constants. Besides appropriate smoothness assumptions, in this paper it is only required that $D(u)\geq C_{D}(u+1)^{m-1}$ for all $u\geq 0$ with some $C_{D} > 0$ and some m>\left\{\begin{array}{ll} 1-\frac{\mu}{\chi[1+\lambda_{0}\|v_0\|_{L^\infty(\Omega)}2^{3}]}~~\mbox{if}~~ N\leq2, % >1+\frac{(N+2-2r)^+}{N+2}~~~~~~\mbox{if}~~ % \frac{N+2}{2}\geq r\geq\frac{N+2}{N}, 1~~~~~~\mbox{if}~~ N\geq3, \end{array}\right. then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where $\lambda_{0}$ is a positive constant which is corresponding to the maximal sobolev regularity. The results of this paper extends the results of Jin (J. Diff. Eqns., 263(9)(2017), 5759-5772), who proved the possibility of boundness of weak solutions, in the case $m>1$ and $N=3$