Global boundedness of weak solutions to a chemotaxis–haptotaxis model with p-Laplacian diffusion

Abstract

In this paper we study a chemotaxis–haptotaxis model with p-Laplacian diffusion and non-flux boundary condition

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\nabla \cdot (|\nabla u|^{p-2}\nabla u)-\chi \nabla \cdot (u\nabla v)-\xi \nabla \cdot (u\nabla w)+\mu u(1-u-w),{} & {} (x,t)\in \Omega \times (0,T),\\&v_{t}=\Delta v-v+u,{} & {} (x,t)\in \Omega \times (0,T),\\&w_{t}=-vw,{} & {} (x,t)\in \Omega \times (0,T), \end{aligned} \right. \end{aligned}$$

where $\Omega \subset \mathbb {R}^n(n\ge 1)$ is a bounded domain with smooth boundary $\partial \Omega$. We prove that when the diffusion parameter is appropriately large, i.e., p larger than some certain value, the global boundedness of the solutions does not depend on the relationship between the coefficient of logistic source term $\mu$ and the coefficient of chemotaxis term $\chi$. And we find that the haptotaxis term does not affect the global boundedness of the solutions, in the other words, our result is consistent with the conclusion with no haptotaxis term (Zhuang et al. in Z Angew Math Phys 72:161, 2021).

A Appendix

1.1 A.1 Preliminaries

In this section, we give the preliminaries used in proving the theorem and lemmas. At first, we introduce a known result on the Neumann heat semigroup. And in order to establish the spatio-temporal integral estimates for the second derivative of v, we need the following lemma.

Lemma A.1

Let $\Omega \subset \mathbb {R}^n(n\ge 1)$ be a bounded domain with smooth boundary, and $1\le p,q\le \infty$.

• (i) [18, Lemma 2.4] If $\frac{n}{2}(\frac{1}{p}-\frac{1}{q})<1$ with $1\le p,q\le \infty$, then there exists $C>0$ such that whenever $\zeta \in C^{2,1}(\bar{\Omega }\times (0,T))\cap C^0(\bar{\Omega }\times [0,T))$ for $T\in (0,\infty ]$ solves

  $$\begin{aligned} \left\{ \begin{aligned}&\zeta _{t}=\Delta \zeta -\zeta +g, \quad{} & {} (x,t)\in \Omega \times (0,T),\\&\partial _{\nu }\zeta =0,{} & {} (x,t)\in \partial \Omega \times (0,T),\\&\zeta (x,0)=\zeta _{0}(x),{} & {} x\in \Omega . \end{aligned} \right. \end{aligned}$$

  (A.1)

  with $g\in C^0(\bar{\Omega }\times [0,T))$ and $\zeta _0\in W^{1,\infty }(\Omega )$, it holds that

  $$\begin{aligned} \Vert \zeta (\cdot ,t)\Vert _{L^q(\Omega )}\le C\Big (\sup _{s\in (0,t)}\Vert g(\cdot ,s)\Vert _{L^p(\Omega )} +\Vert \zeta _0\Vert _{L^\infty (\Omega )}\Big ) \quad { for~each}~t\in (0,T). \end{aligned}$$

• (ii) [18, Lemma 2.4] Assume that $\frac{1}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{q})<1$ with $1\le p,q\le \infty$. Then there exists $C>0$ such that if $\zeta \in C^{2,1}(\bar{\Omega }\times (0,T))\cap C^0(\bar{\Omega }\times [0,T))$ with $T\in (0,\infty ]$ is a solution of (A.1), then

  $$\begin{aligned} \Vert \nabla \zeta (\cdot ,t)\Vert _{L^q(\Omega )}\le C\big (\sup _{s\in (0,t)}\Vert g(\cdot ,s)\Vert _{L^p(\Omega )} +\Vert \nabla \zeta _0\Vert _{L^\infty (\Omega )}\big ) \quad for~any~t\in (0,T). \end{aligned}$$

• (iii) [20, Proposition 2.2] Assume $p,q\in (1,\infty )$. Then there exists $C>0$ such that whenever $\zeta \in L_{loc}^q([0,T);W^{2,p}(\Omega ))\cap W_{loc}^{1,q}([0,T);L^p(\Omega ))$ for $0<T\le \infty$ is the unique strong solution of (A.1) with $g\in L_{loc}^q([0,T);L^p(\Omega ))$, and $\zeta _0\in C^2(\bar{\Omega })$ satisfying $\partial _{\nu }\zeta _0|_{\partial \Omega }=0$, we have

  $$\begin{aligned} \int \limits _{(t-1)_+}^t\Vert (-\Delta +1)\zeta (\cdot ,\tau )\Vert _{L^p(\Omega )}^q d\tau \le C\Big (\sup _{s\in (0,t]}\int \limits _{(s-1)_+}^s\Vert g(\cdot ,\tau )\Vert _{L^p(\Omega )}^qd\tau +\Vert \zeta _0\Vert _{W^{2,p}(\Omega )}^q\Big ) \end{aligned}$$

  for all $t\in (0,T)$.

To treat the p-Laplacian diffusion term in the first equation of (1), we need the following inequalities.

Lemma A.2

Let $\varphi \in C^1(\mathbb {R}^n)$ and $\epsilon >0$. If $1<p<2$, then

$$\begin{aligned} (|\nabla \varphi |^2+\epsilon )^{\frac{p-2}{2}}|\nabla \varphi |^2\ge (|\nabla \varphi |^2+\epsilon )^{\frac{p}{2}}-\epsilon ^{\frac{p}{2}}. \end{aligned}$$

(A.2)

If $p\ge 2$, then

$$\begin{aligned} (|\nabla \varphi |^2+\epsilon )^{\frac{p-2}{2}}|\nabla \varphi |^2\ge |\nabla \varphi |^p. \end{aligned}$$

(A.3)

Proof

For the case of $1<p<2$, we have

$$\begin{aligned} \frac{\epsilon }{|\nabla \varphi |^2+\epsilon } \le \Big (\frac{\epsilon }{|\nabla \varphi |^2+\epsilon }\Big )^{\frac{p}{2}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{|\nabla \varphi |^2}{|\nabla \varphi |^2+\epsilon } \ge 1-\Big (\frac{\epsilon }{|\nabla \varphi |^2+\epsilon }\Big )^{\frac{p}{2}}. \end{aligned}$$

Multiplying by $(|\nabla \varphi |^2+\epsilon )^{\frac{p}{2}}$, we get (A.2).

When $p\ge 2$, it is trivial that

$$\begin{aligned} (|\nabla \varphi |^2+\epsilon )^{\frac{p-2}{2}}| \nabla \varphi |^2\ge (|\nabla \varphi |^2)^{\frac{p-2}{2}}| \nabla \varphi |^2=|\nabla \varphi |^p. \end{aligned}$$

Therefore the lemma is proved. $\square$

The following ordinary differential inequality established in [20] play crucial role in proving the global boundedness of solutions.

Lemma A.3

[20, Lemma 2.4] Assume that $\kappa ,\ell ,c_i(i=1,2,3,4)$ and $y_0$ are positive constants. If $0<\ell <\kappa \le 1$, then there exists $C>0$ such that whenever nonnegative functions $y\in C^1((0,T))\cap C^0([0,T))$ and $h\in C^0([0,T))$ for $T\in (0,\infty ]$ satisfy

$$\begin{aligned} \left\{ \begin{aligned}&y'(t)+c_1y^\kappa (t)\le c_2h(t)+c_3,\quad{} & {} t\in (0,T),\\&y(0)\le y_0 \end{aligned} \right. \end{aligned}$$

(A.4)

and

$$\begin{aligned} \int \limits _{(t-1)_+}^t h(\tau )d\tau \le c_4\big (Y^\ell (t)+1\big ),~~t\in (0,T) \end{aligned}$$

with $Y(t):=\sup _{0<\tau <t}y(\tau ), t\in (0,T)$, it holds that $y\le C$ in (0, T).

In the end of this section, we state an auxiliary statement given in [6, Lemma 2.5]. It is used to prove the global weak solution of (1)

Lemma A.4

[6, Lemma 2.5] For any $\sigma ,\sigma '\in \mathbb {R}^n$, it holds that

$$\begin{aligned} \Big (|\sigma |^{p-2}\sigma -|\sigma '|^{p-2}\sigma '\Big )\cdot (\sigma -\sigma ') \ge \left\{ \begin{aligned}&C(|\sigma |+|\sigma '|)^{p-2}|\sigma -\sigma '|^2,\quad{} & {} \mathrm{if~~} p>1,\\&C|\sigma -\sigma '|^p,\quad{} & {} \mathrm{if~~} p\ge 2 \end{aligned} \right. \end{aligned}$$

(A.5)

with positive constants C depending on p only.