1. Introduction

As a major disease threatening human life safety, cancer has always been the focus of social attention. The combination of mathematical models and various medical data will help people turn cancer research into a quantitative and predictable science. The mathematical modelling of tumour growth and its theoretical research have always been the concern of biologists and mathematicians. Generally speaking, the growth of a tumour usually goes through two stages: vascular phase and vascularization. That is, if there are no blood vessels to provide enough nutrients, the tumour will stop growing when it grows to a certain size. In order to continue to grow, some tumours will secrete a chemical substance, which is called urokinase Plasminogen Activator (uPA), to recruit vascular factors, and to build a vascular network around themselves to meet the growing nutritional needs. At the same time, blood vessels also provide a way for tumours to travel to other parts of the body, which is called metastasis. In recent years, some reaction-diffusion-taxis models are proposed to characterize the process of tumour growth and invasion [Reference Byrne, Chaplain, J.Pettet and Mcelwain1, Reference Chaplain and Anderson3, Reference Chaplain and Lolas4, Reference Stinner and Surulescu17]. In particular, the following model proposed by Chaplain and Lolas [Reference Chaplain and Lolas4] has attracted extensive attention of mathematicians in recent years.

(1.1)\begin{equation} \left\{ \begin{aligned} & u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla\omega)+\mu u(1-u-\omega), \\ & v_t=\Delta v+u-v, \\ & \omega_t={-}v\omega+\eta \omega(1-u-\omega), \end{aligned}\right. \end{equation}

which describes the invasion and diffusion process of solid tumours during the vascular growth stage. In this system, $u$, $v$, $\omega$ represent the cancer cell density, urokinase Plasminogen Activator (uPA) concentration, and the extracellular matrix (ECM) density respectively. $D(u)$ denotes the mobility of cells, the positive constants $\chi$, $\xi$ denote the uPA-mediated chemotaxis, ECM-mediated haptotaxis coefficients respectively, $\mu u(1-u-\omega )$ denotes the proliferation or death of cancer cells. In the second equation, $+u$ represents the production of uPA by cancer cells, $-v$ represents decay of uPA. As for the ECM density $\omega$, it is known that it does not diffuse, therefore, one can omit the random motion. $-v\omega$ denotes the degradation of ECM, $\eta \omega (1-u-\omega )$ represents the remodelling of ECM components.

In the past ten years, this model has been widely studied. When $D(u)\equiv 1$, $\eta =0$, we refer to [Reference Cao5, Reference Marciniak-Czochra and Ptashnyk13, Reference Tao and Wang18, Reference Walker and Webb23] for the study of global existence of bounded solutions, and refer to [Reference Hillen, Painter and Winkler6, Reference Tao and Winkler20, Reference Wang and Ke22] for the study of large time behaviour. While, if the remodelling of ECM is considered, the calculation of the haptotaxis term will bring some essential difficulties. Therefore, the following transformation is introduced to avoid the estimation of the haptotaxis term

\[ \rho=ue^{-\xi\omega}, \]

and the first equation of (1.1) is transformed into

(1.2)\begin{align} e^{\xi\omega}\rho_t-{\rm div}(e^{\xi\omega}\nabla\rho)={-}\chi{\rm div}(\rho e^{\xi\omega}\nabla v) +\xi\rho v\omega e^{\xi\omega}+\rho e^{\xi\omega}(1-\omega-\rho e^{\xi\omega})(\mu -\xi\eta\omega). \end{align}

Noticing that $\omega$ is bounded, then $\|u\|_{L^p}\sim \|\rho \|_{L^p}$. Using this method, the global existence and boundedness of classical solution for any initial datum in two dimensional space is proved [Reference Jin10, Reference Pang and Wang15]. While in three dimensional space, only a small global classical solution is established [Reference Jin11, Reference Pang and Wang16]. In fact, even for the haptotaxis-only system, the global bounded solution for any $\mu,\, \eta >0$ in dimension 3 is still open.

Considering the mechanism of avoiding crowding between individual cells, the mobility of cells should be related to density, so a model with nonlinear diffusion is also very practical, and the most representative one is the porous medium diffusion model. For example, in [Reference Byrne, Chaplain, J.Pettet and Mcelwain1, Reference Byrne, Chaplain, J.Pettet and Mcelwain2], the authors use porous medium diffusion to describe the process of trophoblast cells invading the uterine tissue, that is,

(1.3)\begin{equation} \left\{ \begin{aligned} & u_t=D_u\nabla\cdot(u^{m-1}\nabla u)-\xi\nabla\cdot(u\nabla v)+k_1 u(1-u-\omega), \\ & n_t=\Delta n+k_2nu(1-u)-k_3nv, \\ & v_t=D_v\Delta v+k_4n\omega-k_3nv, \\ & \omega_t={-}\delta n\omega+\eta\omega(1-u-\omega). \end{aligned}\right. \end{equation}

In 2011, Tao and Winkler [Reference Tao and Winkler19] first studied the global solvability of (1.1) by ignoring the remodelling of ECM, that is, for

(1.4)\begin{equation} \left\{ \begin{aligned} & u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla\omega)+\mu u(1-u-\omega), \\ & v_t=\Delta v+u-v, \\ & \omega_t={-}v\omega, \end{aligned}\right. \end{equation}

they proved the global existence of weak solutions when

\[ m> m^*=\begin{cases} \dfrac{2N^2+4N-4}{N(N+4)}, & \text{if} \ N\le 8, \\ \dfrac{2N^2+3N+2-\sqrt{8N(N+1)}}{N(N+1)}, & \text{if} \ N\ge 9. \end{cases} \]

After that, the global existence of weak solutions to this problem began to be widely studied, see for example [Reference Jin7, Reference Li and Lankeit12, Reference Wang21, Reference Zheng24] etc. For this case, from the third equation of (1.4), one observed that

(1.5)\begin{equation} \omega=\omega_0 e^{-\int_0^t v(x,s){\rm d}s}. \end{equation}

At this time, there are two commonly used methods to assist in dealing with the haptotactic term.

Method I: A direct calculation from (1.5) leads to

\[ |\nabla \omega(x,t)|^2\le2|\nabla \omega_0(x)|^2e^{{-}2\int_0^t v(x,s){\rm d}s} +2|w_0(x)|^2e^{{-}2\int_0^tv(x,s){\rm d}s}\Big|\int_0^t\nabla v(x,s){\rm d}s\Big|^2. \]

Therefore,

\begin{align*} \int_\Omega|\nabla \omega(x,t)|^2{\rm d}x & \le C+C\int_\Omega e^{{-}2\int_0^tv(x,s){\rm d}s}\Big|\int_0^t\nabla v(x,s){\rm d}s\Big|^2{\rm d}x\\ & = C-\frac{C}{2}\int_\Omega\nabla e^{{-}2\int_0^tv(x,s){\rm d}s}\cdot\Big(\int_0^t\nabla v(x,s){\rm d}s\Big){\rm d}x\\ & = C+\frac{C}{2}\int_\Omega e^{{-}2\int_0^tv(x,s){\rm d}s}\cdot\Big(\int_0^t\Delta v(x,s){\rm d}s\Big){\rm d}x\\ & = C+\frac{C}{2}\int_\Omega e^{{-}2\int_0^tv(x,s){\rm d}s}\cdot\Big(\int_0^t(v_t+v-u){\rm d}s\Big){\rm d}x\\ & \le C+\frac{C}{2}\int_\Omega e^{{-}2\int_0^tv(x,s){\rm d}s}\cdot\Big(v(x,t)+\int_0^tv(x,s){\rm d}s\Big){\rm d}x \\ & \le \tilde C+\frac{C}{2}\int_\Omega v(x,t){\rm d}x. \end{align*}

Similarly, one can use iteration to sequentially obtain $\|\nabla \omega (\cdot,\,t)\|_{L^4}$, $\|\nabla \omega (\cdot,\,t)\|_{L^6}$, $\|\nabla \omega (\cdot,\,t)\|_{L^{2n}}\cdots$.

Method II: A direct calculation from (1.5) leads to

\begin{align*} -\Delta\omega(x,t)& ={-}\Delta\omega_0e^{-\int_0^t v(x,s){\rm d}s}+2e^{-\int_0^t v(x,s){\rm d}s}\nabla \omega_0\cdot\int_0^t \nabla v(x,s){\rm d}s\nonumber \\ & \quad-\omega_0e^{-\int_0^t v(x,s){\rm d}s}\left|\int_0^t \nabla v(x,s){\rm d}s\right|^2+\omega_0 e^{-\int_0^t v(x,s){\rm d}s}\int_0^t \Delta v(x,s){\rm d}s \\ & ={-}\Delta\omega_0e^{-\int_0^t v(x,s){\rm d}s}-\omega_0e^{-\int_0^t v(x,s){\rm d}s}\left(\frac{\nabla\omega_0}{\omega_0}-\int_0^t \nabla v(x,s){\rm d}s\right)^2\nonumber \\ & \quad +\frac{|\nabla\omega_0|^2}{\omega_0}e^{-\int_0^t v(x,s){\rm d}s}+\omega_0 e^{-\int_0^t v(x,s){\rm d}s}\int_0^t (v_t+v-u){\rm d}s \\ & \le \|\Delta \omega_0\|_{L^\infty}+4\|\nabla\sqrt{\omega_{0}}\|_{L^\infty}^2+\|\omega_{0}\|_{L^\infty}(1+v(x,t)) \\ & \le\|\omega_0\|_{L^\infty}v(x,t)+K \end{align*}

where $K$ only depends on $\omega _0$ [Reference Tao and Winkler19]. Therefore, the above two methods can make the haptotactic term easy to handle, however, the two methods are obviously not suitable for situations where $\eta >0$. In fact, if the remodelling of ECM is considered, that is, the third equation of (1.4) is replaced with

\[ \omega_t={-}v\omega+\eta \omega(1-u-\omega), \]

the problem becomes much more complex since the regularity of $\omega$ completely depends on the regularity of $u$. Actually, due to that the ODE has no regularization effect, the regularity of $\omega$ is greatly reduced by $-\eta u\omega$ in remodelling term. Therefore, although a lot of results for the case $\eta =0$ have been achieved in the past ten years, there is no any progress in the case of $\eta >0$.

In the present paper, we consider the initial and boundary value problem for the system (1.1) with nonlinear diffusion, that is

(1.6)\begin{equation} \begin{cases} u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla\omega)+\mu u(1-u-\omega), & \text{in}\ Q, \\ v_t=\Delta v+u-v, & \text{in}\ Q, \\ \omega_t={-}v\omega+\eta \omega(1-u-\omega), & \text{in}\ Q, \\ \left.\displaystyle\frac{\partial u^m}{\partial{\bf n}}-\chi u\frac{\partial v}{\partial{\bf n}}-\xi u\frac{\partial \omega}{\partial{\bf n}} \right|_{\partial\Omega}=0, \left.\frac{\partial v}{\partial{\bf n}}\right|_{\partial\Omega}=0, & \\ u(x,0)=u_0(x), \quad v(x,0)=v_0(x), \quad \omega(x,0)=\omega_0(x), & x\in\Omega, \end{cases} \end{equation}

where $Q=\Omega \times \mathbb {R}^+$, $m>1$, $\Omega \subset \mathbb {R}^N$ ($N\ge 2$) is a smooth bounded domain. $\chi$, $\xi$, $\mu$, $\eta$ are positive constants.

According to previous research experience for some chemotaxis models, the larger $m$ is, the easier it is to prove the uniform boundedness of the solution, but this experience seems to be invalid for the current model. On the one hand, due to the existence of remodelling term $\eta \omega (1-u-\omega )$, the methods (Method I and Method II) used in the study of equations (1.4) is completely invalid. On the other hand, due to the lack of a good coupling structure similar to the linear diffusion case between the diffusion term and the haptotactic term, the method used for the linear diffusion case (see (1.2)) is also not suitable for the current nonlinear diffusion model. Therefore, we have to try new methods. Fortunately, we found a complex relationship between the haptotactic term and the nonlinear diffusion term. In addition to constructing classical functional such as

\[ F(u,\omega)=\int_\Omega \left(u\ln u+\frac{ |\nabla \omega|^2}{\omega} \right){\rm d}x, \]

we also construct the following new form of functionals

\begin{align*} F_1(u, \omega)& =\int_{\Omega} \left(\frac{1}{m-1}(u+\varepsilon)^m-\xi u\omega \right){\rm d}x,\\ G_k(u, \omega)& =\left(\frac{m}{m-1}\right)^{2k+1}\sum_{i=0}^{2k+1}\frac{(-\frac{m-1}{m}\xi)^iC_{2k+1}^i}{(m-1)(2k+1-i)+1}\\ & \quad \times \int_{\Omega} (u+\varepsilon)^{(m-1)(2k+1-i)+1}\omega^i {\rm d}x, \end{align*}

which allows us to reconstruct the relationship between nonlinear diffusion term and haptotactic term. In particular, by taking the derivative of the above functional, we can use the diffusion term to neutralize the ‘bad’ effect brought by the haptotactic term. That is (see lemma 3.6)

\begin{align*} & \frac{{\rm d}}{{\rm d}t}G_k(u, \omega)+(2k+1)\int_{\Omega}(u+\varepsilon)\left(\frac{m}{m-1}(u+\varepsilon)^{m-1}-\xi\omega\right)^{2k} \\ & \quad \times \left|\nabla\left(\frac{m}{m-1}(u+\varepsilon)^{m-1}-\xi\omega\right)\right|^2{\rm d}x\cdots. \end{align*}

However, this also reduces the original good effect of the diffusion term. Fortunately, we have the logistic term, with the help of this term, we can finally get the $L^p$-norm ($\forall$ $p>1$)uniform estimation of $u$ and the $W^{1,\infty }$-norm estimation of $v$. However, it is hard to get the $L^\infty$-norm estimation of $u$ since the Moser's iteration technique is no longer applicable due to the diffusion term is not working.

Although the haptotactic term caused the main difficulties in the proof, we prove that the chemotaxis still plays a leading role in determining whether the solution can exist globally. More precisely, we prove that the weak solution will exists globally for any $m>\frac {2N}{N+2}$ if $\chi >0$, while for the haptotaxis-only model, that is the case $\chi =0$. We prove that for any $m>1$, the solution always exists globally. This work obviously improves the results in references[Reference Li and Lankeit12, Reference Tao and Winkler19, Reference Wang21, Reference Zheng24], in which, only the special case $\eta =0$ is studied.

In what follows, we give the assumptions of this paper.

\[ (H)\qquad\qquad \begin{cases} u_0, \omega_0 \in L^\infty(\Omega), \nabla\sqrt{\omega_{0}}\in L^2(\Omega),\\ \quad v_0 \in W^{1,\infty}(\Omega)\cap W^{2,p}(\Omega), & \text{for any }p>1, \\ u_0, v_0, \omega_0\ge 0 & . \end{cases} \]

We state the main results as follows.

Theorem 1.1 Assume that (H) holds, $\mu >0$ and $m>\frac {2N}{N+2}$. Then the problem (1.6) admits a global nonnegative weak solution $(u,\, v,\, \omega )\in \mathcal {X}_1\times \mathcal {X}_2\times \mathcal {X}_3$. In particular, $u,\, v,\, \omega$ are bounded uniformly in the following sense

(1.7)\begin{align} \sup_{0< t<\infty}\left(\|v({\cdot}, t)\|_{W^{1,\infty}}+\|\omega({\cdot}, t)\|_{L^{\infty}}+\|u({\cdot}, t)\|_{L^{r}}\right)\le C_r,\quad \text{for any $r>1$ }, \end{align}

where $C_r$ only depends on $r$, $\chi$, $\xi$, $m$, $\mu$, $\eta$, $u_0$, $v_0$, $\omega _0$, $\Omega$. Here

\begin{align*} \mathcal{X}_1& =\Big\{u\in L^\infty(\mathbb{R}^+; L^r(\Omega)) \ \text{ for any}\ r>1; \nabla u^m\in L_{loc}^p(\mathbb{R}^+; L^p(\Omega)),\\ & \quad \qquad u_t\in L_{loc}^2(\mathbb{R}^+; W^{{-}1,p}(\Omega)), \\ & \quad \text{ for any}\ p\in (1,2), \nabla u^{m-\frac12}\in L_{loc}^2(\mathbb{R}^+; L^2(\Omega)) \Big\}; \\ \mathcal{X}_2& =\{v\in L^\infty(\mathbb{R}^+; W^{1,\infty}(\Omega)); v_t, D^2v\in L_{loc}^p(\mathbb{R}^+; L^p(\Omega)) \text{ for any}\ p>1 \}; \\ \mathcal{X}_3& =\{\omega\in L^\infty(Q);\nabla\sqrt\omega \in L_{loc}^\infty(\mathbb{R}^+; L^2(\Omega)), \sqrt u\nabla\sqrt\omega\in L_{loc}^2(\mathbb{R}^+; L^2(\Omega)), \\ & \quad\omega_t\in L_{loc}^r(\mathbb{R}^+; L^r(\Omega)) \ \text{ for any}\ r>1 \}. \end{align*}

Although the haptotactic term caused the main difficulties in the proof, the requirement of this index $m>\frac {N+2}{2N}$ was still caused by the chemotactic term. In fact, if $\chi =0$, we have the global existence result for any slow diffusion case $m>1$.

Theorem 1.2 Assume $\chi =0$, $\mu >0$, $m>1$ and (H) holds. Then the problem (1.6) admits a global nonnegative weak solution $(u,\, v,\, \omega )\in \mathcal {X}_1\times \mathcal {X}_2\times \mathcal {X}_3$. In particular, $u,\, v,\, \omega$ are bounded uniformly in the following sense

(1.8)\begin{equation} \sup_{0< t<\infty}\left(\|v({\cdot}, t)\|_{W^{1,\infty}}+\|\omega({\cdot}, t)\|_{L^{\infty}}+\|u({\cdot}, t)\|_{L^{r}}\right)\le C_r,\quad \text{for any }r>1, \end{equation}

where $C_r$ only depends on $r$, $\chi$, $\xi$, $m$, $\mu$, $\eta$, $u_0$, $v_0$, $\omega _0$, $\Omega$.