Skip to main content
Log in

Stabilization to a cancer invasion model with remodeling mechanism and slow diffusion

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we consider the stability of solutions to a class of cancer invasion model. This kind of model was first proposed by Chaplain and Lolas in 2005. In fact, they proposed two kinds of tumor invasion models in 2005 (Chaplain and Lolas in Math Models Methods Appl Sci 15(11):1685–1734, 2005) and 2006 (Chaplain and Lolas in Netw Heterog Media 1(3):399–439, 2006), respectively, which have similar structures, but their solutions have different properties. For convenience of distinction, we call them Model I and Model II, respectively. A common feature of the two models is that they both consider the remodeling of extracellular matrix. For the research on the stability of solutions of the two models, there are only the three papers (Hillen in Math Models Methods Appl Sci 23(01):165–198, 2013; Tao in SIAM J Math Anal 47(6): 4229–4250, 2015; Wang in J Differ Equ 260(9):6960–6988, 2016) on Model II by removing the remodeling of extracellular matrix, and the paper (Jin in Nonlinearity 33(10):5049–5079, 2020) on Model I. However, as far as we know, although the existence of solutions of this kind of model with porous medium diffusion has achieved fruitful results, there are no relevant results on stability even without considering the remodeling effect of extracellular matrix. In the present paper, we consider the stability of this kind of model with nonlinear diffusion, we find that the simultaneous emergence of haptotaxis, nonlinear diffusion and remodeling effect does bring essential difficulties to the study of stability, such as the method of constructing Lyapunov functional is no longer applicable. In this paper, we use some detailed analytical techniques to prove the global asymptotic stability of bounded solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15(11), 1685–1734 (2005)

    Article  MathSciNet  Google Scholar 

  2. Plekhanova, O., et al.: Urokinase plasminogen activator augments cell proliferation and neointima formation in injured arteries via proteolytic mechanisms. Atherosclerosis 159(2), 297–306 (2001)

    Article  Google Scholar 

  3. Aguirre-Ghiso, J.A., Liu, D., Mignatti, A., Kovalski, K., Ossowski, L.: Urokinase receptor and fibronectin regulate the ERKMAPK to p38MAPK activity ratios that determine carcinoma cell proliferation or dormancy in vivo. Mol. Biol. Cell 12(4), 863–879 (2001)

    Article  Google Scholar 

  4. Ossowski, L., Aguirre-Ghiso, J.A.: Urokinase receptor and integrin partnership: coordination of signaling for cell adhesion, migration and growth. Curr. Opin. Cell Biol. 12(5), 613–620 (2000)

    Article  Google Scholar 

  5. Andreasen, P.A., et al.: The urokinase-type plasminogen activator system in cancer metastasis: a review. Int. J. Cancer 72(1), 1–22 (1997)

    Article  MathSciNet  Google Scholar 

  6. Andreasen, P.A., Egelund, R., Petersen, H.H.: The plasminogen activation system in tumor growth, invasion, and metastasis. Cell. Mol. Life Sci. 57(1), 25–40 (2000)

    Article  Google Scholar 

  7. Andreasen, P.A., et al.: Receptor-mediated endocytosis of plasminogen activators and activator/inhibitor complexes. FEBS Lett. 338(3), 239–45 (1994)

    Article  Google Scholar 

  8. Szymańska, Z., et al.: Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math. Models Methods Appl. Sci. 19(2), 257–281 (2009)

    Article  MathSciNet  Google Scholar 

  9. Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Netw. Heterog. Media 1(3), 399–439 (2006)

    Article  MathSciNet  Google Scholar 

  10. Jin, C.: Global solvability and stabilization to a cancer invasion model with remodelling of ECM. Nonlinearity 33(10), 5049–5079 (2020)

    Article  MathSciNet  Google Scholar 

  11. Tao, Y., Winkler, M.: A chemotaxis–haptotaxis model: the roles of nonlinear diffusion and logistic Source. SIAM J. Math. Anal. 43(2), 685–704 (2011)

    Article  MathSciNet  Google Scholar 

  12. Li, Y., Lankeit, J.: Boundedness in a chemotaxis–haptotaxis model with nonlinear diffusion. Nonlinearity 29(5), 1564 (2016)

    Article  MathSciNet  Google Scholar 

  13. Wang, Y.: Boundedness in the higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. J. Differ. Equ. 260(2), 1975–1989 (2016)

    Article  MathSciNet  Google Scholar 

  14. Zheng, J.: Boundedness of solutions to a quasilinear higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. Discrete Contin. Dyn. Syst. 37(1), 627–643 (2017)

    Article  MathSciNet  Google Scholar 

  15. Pang, P.Y.H., Wang, Y.: Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling. Math. Models Methods Appl. Sci. 28(11), 2211–2235 (2018)

    Article  MathSciNet  Google Scholar 

  16. Pang, P.Y.H., Wang, Y.: Global existence of a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant. J. Differ. Equ. 263(2), 1269–1292 (2017)

    Article  MathSciNet  Google Scholar 

  17. Jin, C.: Global classical solution and boundedness to a chemotaxis–haptotaxis model with re-establishment mechanisms. Bull. Lond. Math. Soc. 50(4), 598–618 (2018)

    Article  MathSciNet  Google Scholar 

  18. Hillen, T., Painter, K.J., Winkler, M.: Convergence of a cancer invasion model to a logistic chemotaxis model. Math. Models Methods Appl. Sci. 23(01), 165–198 (2013)

    Article  MathSciNet  Google Scholar 

  19. Wang, Y., Ke, Y.: Large time behavior of solution to a fully parabolic chemotaxis–haptotaxis model in higher dimensions. J. Differ. Equ. 260(9), 6960–6988 (2016)

    Article  MathSciNet  Google Scholar 

  20. Tao, Y., Winkler, M.: Large time behavior in a multidimensional chemotaxis–haptotaxis Model with slow signal diffusion. SIAM J. Math. Anal. 47(6), 4229–4250 (2015)

    Article  MathSciNet  Google Scholar 

  21. Jin, C.: Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread. J. Differ. Equ. 269(4), 3987–4021 (2020)

    Article  MathSciNet  Google Scholar 

  22. Jin, C.: Global bounded weak solutions and asymptotic behavior to a chemotaxis–Stokes model with non-Newtonian filtration slow diffusion. J. Differ. Equ. 287, 148–184 (2021)

    Article  MathSciNet  Google Scholar 

  23. Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215(1), 52–107 (2004)

    Article  MathSciNet  Google Scholar 

  24. Tao, Y.: Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source. J. Math. Anal. Appl. 354(1), 60–69 (2008)

    Article  MathSciNet  Google Scholar 

  25. Chen, X., Jüngel, A., Liu, J.-G.: A note on Aubin–Lions–Dubinskiĭ lemmas. Acta Appl. Math. 133(1), 33–43 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which made some meaningful changes in this revision.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Danqing Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by NSFC (11871230), Guangdong Basic and Applied Basic Research Foundation (2020B1515310013).

A Appendix

A Appendix

The proof of Lemma 3.1 is based on energy estimates in a regularized system. More precisely, in order to solve the equation of u, we replace the nonlinear slow diffusion term with a non-degenerate one, that is to say, we work on the approximate problems given by

$$\begin{aligned} \left\{ \begin{aligned}&{u_{\varepsilon t}} = \Delta {(u_\varepsilon ^2 + \varepsilon )^{\frac{{m - 1}}{2}}}{u_\varepsilon } - \chi \nabla \cdot ({u_\varepsilon }\nabla {v_\varepsilon }) - \xi \nabla \cdot \left( {{u_\varepsilon }\nabla {w_\varepsilon }} \right) - \mu {u_\varepsilon }(1 - {u_\varepsilon }) + \beta {u_\varepsilon }{v_\varepsilon },\quad in\;Q,\\&\quad {v_{\varepsilon t}} = D\Delta {v_\varepsilon } + {u_\varepsilon } - {u_\varepsilon }{v_\varepsilon },\quad in\;Q,\\&\quad {w_{\varepsilon t}} = - \delta {v_\varepsilon }{w_\varepsilon } + \eta {w_\varepsilon }\left( {1 - {w_\varepsilon }} \right) ,\quad in\;Q,\\&\quad {\left. {{{\left. {\frac{{\partial u_\varepsilon ^{}}}{{\partial n}}} \right| }_{\partial n}} = {{\left. {\frac{{\partial {v_\varepsilon }}}{{\partial n}}} \right| }_{\partial n}} = \frac{{\partial {w_\varepsilon }}}{{\partial n}}} \right| _{\partial n}} = 0,\\&\quad {u_\varepsilon }(x,0) = {u_{\varepsilon 0}}(x),\quad {v_\varepsilon }(x,0) = {v_{\varepsilon 0}}(x),\quad {w_\varepsilon }(x,0) = {w_{\varepsilon 0}}(x),\quad x \in \Omega , \end{aligned}\right. \end{aligned}$$
(A.1)

where \({u_{\varepsilon 0}},{v_{\varepsilon 0}},{w_{\varepsilon 0}}\) satisfy

$$\begin{aligned} \left\{ \begin{aligned}&{u_{\varepsilon 0}},{v_{\varepsilon 0}},{w_{\varepsilon 0}} \in {C^{2 + \alpha }}(\bar{\Omega }),\;\alpha \in (0,1), \\&\quad {u_{\varepsilon 0}},{v_{\varepsilon 0}},{w_{\varepsilon 0}} \ge 0,\;{\left\| {{v_{\varepsilon 0}}} \right\| _{{L^\infty }}} \le {\left\| {{v_0}} \right\| _{{L^\infty }}},\;{\left\| {{w_{\varepsilon 0}}} \right\| _{{L^\infty }}} \le {\left\| {{w_0}} \right\| _{{L^\infty }}}, \\&\quad {\left. {\frac{{\partial {u_{\varepsilon 0}}}}{{\partial n}}} \right| _{\partial \Omega }} = {\left. {\frac{{\partial {v_{\varepsilon 0}}}}{{\partial n}}} \right| _{\partial \Omega }} = {\left. {\frac{{\partial {w_{\varepsilon 0}}}}{{\partial n}}} \right| _{\partial \Omega }} = 0, \\&\quad {u_{\varepsilon 0}} \rightarrow {u_0}\;in\;{L^p}\;for\;any\;p \in (1,\infty ),\;{v_{\varepsilon 0}} \rightarrow {v_0},\;{w_{\varepsilon 0}} \rightarrow {w_0},\quad uniformly. \\ \end{aligned} \right. , \end{aligned}$$
(A.2)

It is not difficult to obtain

$$\begin{aligned} 0 \le \left\| {{v_\varepsilon }} \right\| \le \max \{ 1,{\left\| {{v_{\varepsilon 0}}} \right\| _{{L^\infty }}}\} \le \max \{ 1,{\left\| {{v_0}} \right\| _{{L^\infty }}}\} , \end{aligned}$$
(A.3)

by comparison principle. By the third equation of (A.1), it is easy to see that

$$\begin{aligned} 0 \le \left\| {{w_\varepsilon }} \right\| \le \max \{ 1,{\left\| {{w_{\varepsilon 0}}} \right\| _{{L^\infty }}}\} \le \max \{ 1,{\left\| {{w_0}} \right\| _{{L^\infty }}}\} . \end{aligned}$$
(A.4)

We must point out that, \({v_\varepsilon }\) and \({w_\varepsilon }\) are bounded uniformly.

Using fixed point theory, or similar to the study of the chemotaxis model [15, 24], it is not difficult to obtain the following local existence result of classical solution to the problem (A.1) for any \(\varepsilon > 0\).

Lemma A.1

Assume that \({u_{\varepsilon 0}},\;{v_{\varepsilon 0}},\;{w_{\varepsilon 0}}\) satisfy (A.2). Then for any \(\varepsilon >0\), there exists \({T_{\max }} \in (0, + \infty ]\) such that the problem (A.1) admits a unique classical solution \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon }) \in {C^{2 + \alpha ,1 + \frac{\alpha }{2}}}(\Omega \times [0,{T_{\max }}))\) with

$$\begin{aligned} {u_\varepsilon } \ge 0,\;{v_\varepsilon } \ge 0,\;{w_\varepsilon } \ge 0\quad for\;all\;(x,t) \in \Omega \times (0,{T_{\max }}) \end{aligned}$$

such that either \({T_{\max }} = \infty \), or

$$\begin{aligned} \mathop {\lim }\limits _{t \nearrow {T_{\max }}} ({\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}} + {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{W^{1,\infty }}}} + {\left\| {{w_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}}) = + \infty . \end{aligned}$$
(A.5)

In order to verify that the problem (A.1) admits a unique global solution, we may assume \({T_{\max }} < \infty \), our purpose is to show the boundedness of \({\left\| {u( \cdot ,t)} \right\| _{{L^\infty }}} + {\left\| {v( \cdot ,t)} \right\| _{{W^{1,\infty }}}} + {\left\| {w( \cdot ,t)} \right\| _{{L^\infty }}}\) for any \(t \in [0,{T_{\max }}]\).

It is not difficult to obtain the following lemma.

Lemma A.2

Assume (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have

$$\begin{aligned}&\sup _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^1}}} + \int \limits _0^T {\int \limits _\Omega {u_\varepsilon ^2} } {\mathrm{d}}x{\mathrm{d}}t \le {C_T}, \end{aligned}$$
(A.6)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{H^1}}} + \int \limits _0^T {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{H^2}}^2} {\mathrm{d}}t \le {C_T}, \end{aligned}$$
(A.7)
$$\begin{aligned}&\sup _{t \in (0,T)} {\left\| {{w_{\varepsilon t}}} \right\| _{{L^\infty }}} \le C, \end{aligned}$$
(A.8)

where the constants \({C_T}\) at most depend on \(\chi ,\xi ,\mu ,\beta ,D,{u_0},{v_0},{w_0},T\) and \(\left| \Omega \right| \), C at most depend on \(\chi ,\xi ,\mu ,\beta ,D,{u_0},{v_0},{w_0}, \left| \Omega \right| \), all of them are independent of \(\varepsilon \).

Proof

By a direct integration for the first equation of (A.1), noticing that \({{v_\varepsilon }}\) is bounded, and applying Young’s inequality, we have

$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{u_\varepsilon }{\hbox {d}}x}&= \mu \int \limits _\Omega {{u_\varepsilon }{\hbox {d}}x} - \mu \int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} + \beta \int \limits _\Omega {{u_\varepsilon }{v_\varepsilon }{\hbox {d}}x}\\&\le - \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} + {C_1} , \end{aligned}$$

it means

$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{u_\varepsilon }{\hbox {d}}x} + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} \le {C_1}, \end{aligned}$$
(A.9)

then (A.6) is obtained by a direct integration.

Multiplying the second equation of (A.1) by \({v_\varepsilon }\) and \(\Delta {v_\varepsilon }\), respectively, we have

$$\begin{aligned} \frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x} + D\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{1}{2}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x}&= \int \limits _\Omega {{u_\varepsilon }(1 - {v_\varepsilon }){v_\varepsilon }{\hbox {d}}x} + \frac{1}{2}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x}\\&\le {C_2}\int \limits _\Omega {u_\varepsilon ^{}{\hbox {d}}x} + \frac{1}{2}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + D\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + \int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} {\text { }}&= - \int \limits _\Omega {({u_\varepsilon }(1 - {v_\varepsilon }) + {v_\varepsilon })\Delta {v_\varepsilon }{\hbox {d}}x}\\&\le \frac{D}{2}\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_3}\left( {1 + \int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} } \right) , \end{aligned}$$

adding the above two formulas, since \({v_\varepsilon }\) is bounded, and using (A.6), we obtain (A.7) by a direct integration.

By the third equation of (A.1), since \({{v_\varepsilon }},{{w_\varepsilon }}\) are bounded, we easily see that

$$\begin{aligned} {\left\| {{w_{\varepsilon t}}} \right\| _{{L^\infty }}} \le \delta {\left\| {{v_\varepsilon }{w_\varepsilon }} \right\| _{{L^\infty }}} + \eta {\left\| {{w_\varepsilon }(1 - {w_\varepsilon })} \right\| _{{L^\infty }}} \le {C_4}, \end{aligned}$$

then (A.8) is achieved. \(\square \)

Lemma A.3

Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have

$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^p}} {\mathrm{d}}x \le C\mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {\frac{{{{\left| {\nabla {w_\varepsilon }} \right| }^p}}}{{{w_\varepsilon }}}{\mathrm{d}}x} \le Cp\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^p}{\mathrm{d}}x} {\mathrm{d}}s + {C_T}} ,\;for\;any\;p \ge 2, \end{aligned}$$
(A.10)
$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}} {\mathrm{d}}x \le C\left( {\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}\frac{{{{\left| {\nabla {w_\varepsilon }} \right| }^2}}}{{{w_\varepsilon }}}{\mathrm{d}}x{\mathrm{d}}t} } + \int \limits _0^T {\int \limits _\Omega {\frac{{{{\left| {\nabla {w_\varepsilon }} \right| }^4}}}{{{w_\varepsilon }}}{\mathrm{d}}x} } {\mathrm{d}}t} \right) + {C_T}, \end{aligned}$$
(A.11)

where C at most depend on \(\delta ,\eta ,{v_0},{w_0}\), \({C_T}\) also depend on T, all of them are independent of \(\varepsilon \).

Proof

The proof is similar to Lemma 3.5, so we omit it here. \(\square \)

Lemma A.4

Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\), \(m>1\). Then, for any \(q=1,2,3, \cdots \), we have

$$\begin{aligned}&\sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^q} - 1}}}^{2{m^q} - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{m + 2{m^q} - 4}} {{\left| {\nabla {u_\varepsilon }} \right| }^2}{\mathrm{d}}x + \int \limits _\Omega {u_\varepsilon ^{2{m^q}}} {\mathrm{d}}x} \right) } {\mathrm{d}}s \le {C_T}, \end{aligned}$$
(A.12)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{W^{1,\infty }}}} + \int \limits _0^T {\left( {\left\| {{v_\varepsilon }} \right\| _{{W^{2,2{m^q}}}}^{2{m^q}} + \left\| {{v_{\varepsilon t}}} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} \right) } {\mathrm{d}}s \le {C_T},\quad for\;any\;q > 1. \end{aligned}$$
(A.13)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in [0,T]} \left( {\int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^p}} {\mathrm{d}}x + \int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}} } \right) {\mathrm{d}}x \le {C_T},\quad for\;any\;p \ge 2 \end{aligned}$$
(A.14)

where \({C_T}\) depends on q and T and is independent of \(\varepsilon \).

Proof

Multiplying the first equation of (A.1) by \(u_\varepsilon ^r\) for any \(r>0\), and integrating it over \(\Omega \). Since \({v_\varepsilon }\) is bounded, we infer that

$$\begin{aligned}&\frac{1}{{r + 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + r\int \limits _\Omega {u_\varepsilon ^{m + r - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \mu \int \limits _\Omega {u_\varepsilon ^{r + 2}{\hbox {d}}x}\\&\quad \le r\chi \int \limits _\Omega {u_\varepsilon ^r\nabla {u_\varepsilon }\nabla {v_\varepsilon }{\hbox {d}}x} + r\xi \int \limits _\Omega {u_\varepsilon ^r\nabla {u_\varepsilon }\nabla {w_\varepsilon }{\hbox {d}}x} + \mu \int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + \beta \int \limits _\Omega {u_\varepsilon ^{r + 1}{v_\varepsilon }{\hbox {d}}x}\\&\quad \le \frac{r}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{13}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{14}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{r + 2}{\hbox {d}}x} +{C_{15}}, \end{aligned}$$

which means

$$\begin{aligned}&\frac{1}{{r + 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + \frac{r}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{r + 2}{\hbox {d}}x} \nonumber \\&\quad \le {C_{13}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{14}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{15}}. \end{aligned}$$
(A.15)

By Lemma 2.1, for any \(q=1,2, \cdots \), we have

$$\begin{aligned} \int \limits _0^T {\left\| {\nabla {v_\varepsilon }( \cdot ,t)} \right\| _{{L^{4{m^q}}}}^{4{m^q}}{\hbox {d}}s} {\text { }}&\le C\mathop {\sup }\limits _{t \in (0,T)} \left\| {{v_\varepsilon }} \right\| _{{L^\infty }}^{2{m^q}}\int \limits _0^T {\left\| {\Delta {v_\varepsilon }( \cdot ,s)} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} {\hbox {d}}s + C'\int \limits _0^T {\left\| {{v_\varepsilon }} \right\| _{{L^\infty }}^{4{m^q}}} {\hbox {d}}s \nonumber \\&\le {{C_T}} + {C''}\int \limits _0^T {\left\| {\Delta {v_\varepsilon }( \cdot ,s)} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} {\hbox {d}}s, \end{aligned}$$
(A.16)

where \( {{C_T}}\) depends on \(m,q,\left| \Omega \right| ,{\left\| {{v_0}} \right\| _{{L^\infty }}},T\). Taking \(r=2(m-1)\) in (A.15), then by (A.7), (A.10) and (A.16) with \(q=0\), we infer that

$$\begin{aligned}&\frac{1}{{2m - 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{2m - 1}{\hbox {d}}x} + (m - 1)\int \limits _\Omega {u_\varepsilon ^{3m - 4}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} \nonumber \\&\quad \le (m - 1)\int \limits _\Omega {u_\varepsilon ^m{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {u_\varepsilon ^m{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{17}} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_{17}}\int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_{17}} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + \left( {{C_{19}}\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} } {\hbox {d}}t + {C_T}} \right) + {C_{17}} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_T}\int \limits _0^T {\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}{\hbox {d}}x} } {\hbox {d}}t + {C_T} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_T}, \end{aligned}$$
(A.17)

then by a direct integration, and using (A.16) with \(q=0\), we have

$$\begin{aligned} \sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2m - 1}}}^{2m - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{3m - 4}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} } \right) {\hbox {d}}s} \le {C_T}. \end{aligned}$$
(A.18)

Next, we use recursive method to prove (A.12). Assume that

$$\begin{aligned} \sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^q} - 1}}}^{2{m^q} - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{m + 2{m^q} - 4}} {{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x + \int \limits _\Omega {u_\varepsilon ^{2{m^q}}} {\hbox {d}}x} \right) } {\hbox {d}}s \le {C_T}. \end{aligned}$$
(A.19)

By the second equation of (A.1), we see that

$$\begin{aligned} {v_{\varepsilon t}} - D\Delta {v_\varepsilon } + {v_\varepsilon } = {u_\varepsilon }(1 - {v_\varepsilon }) + {v_\varepsilon }, \end{aligned}$$

using the \(L^p\) theory of linear parabolic equations, we obtain

$$\begin{aligned} \int \limits _0^T {\left( {\left\| {{v_\varepsilon }} \right\| _{{W^{2,2{m^q}}}}^{2{m^q}} + \left\| {{v_{\varepsilon t}}} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} \right) {\hbox {d}}s} \le C\int \limits _0^T {\left\| {{u_\varepsilon }} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} {\hbox {d}}s + {C_T}\left\| {{v_{\varepsilon 0}}} \right\| _{{W^{2,2{m^q}}}}^{2{m^q}} \le {C_T} . \end{aligned}$$
(A.20)

Taking \(r = 2{m^{q + 1}} - 2\) in (A.15), and using (A.10) and (A.16), we see that

$$\begin{aligned}&\frac{1}{{2{m^{q + 1}} - 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}} - 1}{\hbox {d}}x} + \frac{{2{m^{q + 1}} - 2}}{2}\int \limits _\Omega {u_\varepsilon ^{m + 2{m^{q + 1}} - 4}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} {\hbox {d}}x + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x}\\&\quad \le {C_{20}}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}} - m}{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{21}}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}} - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + {C_{24}}\int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + {C_{25}}\left( {\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} } + {C_T}} \right) + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + {C_T}\left( {\int \limits _0^T {\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^{2{m^q}}}{\hbox {d}}x} } + 1} \right) + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_T}, \end{aligned}$$

then by a direct integration and using (A.16) and (A.20), we have

$$\begin{aligned} \sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^{q + 1}} - 1}}}^{2{m^{q + 1}} - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{m + 2{m^{q + 1}} - 4}} {{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x + \int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}} {\hbox {d}}x} \right) } {\hbox {d}}s \le {C_T}. \end{aligned}$$

Then, (A.19) is valid for any \(q=1,2,3, \cdots \).

By (A.19) and (A.20), we see that there exists a large enough \(q'\) such that \(2{m^{q'}} \ge 10\). Then, by t-anisotropic embedding theorem, we have

$$\begin{aligned} {\left\| {{v_\varepsilon }} \right\| _{{C^{\frac{3}{2},\frac{3}{4}}}({Q_T})}} \le C{\left\| {{v_\varepsilon }} \right\| _{W_{10}^{2,1}({Q_T})}} \le {C_T}, \end{aligned}$$

it means

$$\begin{aligned} \sup _{t \in (0,T )} {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{W^{1,\infty }}}} \le {C_T}. \end{aligned}$$

Combining this with (A.20), we obtain (A.13).

Using (A.10) and (A.13), we obtain

$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^p}} \mathrm{d}x\mathrm{d}x \le {C_T},\quad for\;any\;p \ge 2 \end{aligned}$$

Combining this with (A.11) and (A.13), we obtain (A.14). The lemma is proved. \(\square \)

Next, we estimate the \({\left\| {{u_\varepsilon }} \right\| _{{L^\infty }}}\) by Moser’s iterative technique.

Lemma A.5

Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have

$$\begin{aligned}&\sup _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}} \le {C_T}, \end{aligned}$$
(A.21)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} \int \limits _\Omega {{u_\varepsilon }\ln {u_\varepsilon }} \mathrm{d}x + \int \limits _0^T {\int \limits _\Omega {u_\varepsilon ^{m - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} } \mathrm{d}x\mathrm{d}s \le {C_T}, \end{aligned}$$
(A.22)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} {\left\| v \right\| _{W_{p'}^{2,1}({Q_T})}} \le {C_T}\quad for\;any\;p' > 1, \end{aligned}$$
(A.23)

where \({C_T}\) depends on \(m,\chi ,\xi ,\mu ,\beta ,\left| \Omega \right| ,{w_\varepsilon },T \), are independent of \(\varepsilon \).

Proof

In this section, all the constants \({C_i}(i = 25,26,27,28 )\) are also independent of r. We just consider it in dimension 2, the case in dimension 3 is almost the same, so we omit it. Multiplying the first equation by \(ru_\varepsilon ^{r - 1}\) with \(r \ge 2m\), and integrating it over \(\Omega \), by (A.3), (A.13) and (A.14), we see that

$$\begin{aligned}&\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^rdx} + r(r - 1)\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \mu r\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^rdx} \nonumber \\&\quad \le \chi r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r - 1}\nabla {u_\varepsilon }\nabla {v_\varepsilon }{\hbox {d}}x} + \xi r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r - 1}\nabla {u_\varepsilon }\nabla {w_\varepsilon }{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^r(\mu r + \beta r{v_\varepsilon } + 1){\hbox {d}}x} \nonumber \\&\quad \le \frac{{r(r - 1)}}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{26}}r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r + 1 - m}{\hbox {d}}x}+ {C_{27}}r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r + 1 - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} \nonumber \\&\qquad + {C_{28}}r\int \limits _\Omega {u_\varepsilon ^rdx} \nonumber \\&\quad \le \frac{{r(r - 1)}}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{29}}r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r + 1 - m}{\hbox {d}}x}+ {C_T}r(r - 1){\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}{\hbox {d}}x} } \right) ^{\frac{1}{2}}} \nonumber \\&\qquad + \frac{{\mu r}}{2}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} \nonumber \\&\quad \le \frac{{r(r - 1)}}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_T}{r^2}{\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}{\hbox {d}}x} } \right) ^{\frac{1}{2}}} + \frac{{\mu r}}{2}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x}, \end{aligned}$$
(A.24)

it means

$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^r{\hbox {d}}x} + \int \limits _\Omega {{{\left| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right| }^2}} {\hbox {d}}x + \int \limits _\Omega {u_\varepsilon ^rdx} \le {C_T}{r^2}{\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}} {\hbox {d}}x} \right) ^{\frac{1}{2}}}. \end{aligned}$$
(A.25)

It is not difficult to see that by Lemma 2.1

$$\begin{aligned} {C_T}{r^2}{\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}{\hbox {d}}x} } \right) ^{\frac{1}{2}}}&= {C_T}{r^2}\left\| {u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^{\frac{{4(r + 1 - m)}}{{r + m - 1}}}}}^{\frac{{2(r + 1 - m)}}{{r + m - 1}}} \\&\le {C_T}{r^2}\left\| {u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^{\frac{r}{{r + m - 1}}}}}^{\frac{r}{{2(r + m - 1)}}}\left\| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^2}}^{\frac{{3r + 4(1 - m)}}{{2(r + m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{\frac{r}{2}}^{r + 1 - m} \\&\le \frac{1}{2}\left\| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^2}}^2 + {C_T}{r^{\frac{{8r + 8(m - 1)}}{{r + 8(m - 1)}}}}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r \cdot \frac{{r + m - 1}}{{r + 8(m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r + 1 - m} \\&\le \frac{1}{2}\left\| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^2}}^2 + {C_T}{r^8}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r \cdot \frac{{r + m - 1}}{{r + 8(m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r + 1 - m}, \end{aligned}$$

then substituting the above inequality into (A.25), we have

$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^r{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^r{\hbox {d}}x} \le {C_T}{r^8}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r \cdot \frac{{r + m - 1}}{{r + 8(m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r + 1 - m}. \end{aligned}$$

By (A.12), there exists \({q'}>0\) such that \(2{m^{q'}} - 1 \ge 2m\) and

$$\begin{aligned} \mathop {\sup }\limits _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^{q'}} - 1}}}} \le C. \end{aligned}$$

Taking \({r_j} = 2{r_{j - 1}}\) with \({r_0} = 2{m^{q'}} - 1\), \({M_j} = \max \{ \mathop {\sup }\limits _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{{r_j}}}}},{\left\| {{u_{\varepsilon 0}}} \right\| _{{L^\infty }}},1\} \), then by a direct calculation, we obtain

$$\begin{aligned} {M_j} \le C_T^{\frac{1}{{{r_j}}}}r_j^{\frac{8}{{{r_j}}}}{M_{j - 1}} = C_T^{\frac{1}{{{2^j}{r_0}}}}r_j^{\frac{8}{{{2^j}{r_0}}}}{M_{j - 1}} = C_T^{\frac{1}{{{2^j}{r_0}}}} \cdot {r_0}^{\frac{8}{{{2^j}{r_0}}}} \cdot {2^{\frac{{8j}}{{{2^j}{r_0}}}}}{M_{j - 1}}, \end{aligned}$$

then through the iterative process, we have

$$\begin{aligned} {M_n} \le C_T^{\frac{1}{{{r_0}}}\sum \limits _{j = 1}^n {\frac{1}{{{2^j}}}} }{r_0}^{\frac{8}{{{r_0}}}\sum \limits _{j = 1}^n {\frac{1}{{{2^j}}}} }{2^{\frac{8}{{{r_0}}}\sum \limits _{j = 1}^n {\frac{j}{{{2^j}}}} }}{M_0}. \end{aligned}$$

Let \(n \rightarrow \infty \), it is not difficult to see that \({\sum \limits _{j = 1}^n {\frac{1}{{{2^{j }}}}} }\) and \({\sum \limits _{j = 1}^n {\frac{{j}}{{{2^{j }}}}} }\) are convergent, so we have

$$\begin{aligned} \sup _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}} \le {C_T}. \end{aligned}$$

Multiplying \(1+\ln {u_\varepsilon }\) to the first equation of (A.1), using (A.13), (A.14), (A.21), we have

$$\begin{aligned}&\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{u_\varepsilon }\ln {u_\varepsilon }} {\hbox {d}}x + m\int \limits _\Omega {u_\varepsilon ^{m - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} {\hbox {d}}x + \mu \int \limits _\Omega {u_\varepsilon ^2(1 + \ln {u_\varepsilon })} {\hbox {d}}x\\&\quad \le - \chi \int \limits _\Omega {{u_\varepsilon }\Delta {v_\varepsilon }} {\hbox {d}}x - \xi \int \limits _\Omega {{u_\varepsilon }\Delta {w_\varepsilon }} {\hbox {d}}x + \int \limits _\Omega {\left( {\beta {v_\varepsilon } + \mu } \right) u_\varepsilon ^{}} \left( {1 + \ln {u_\varepsilon }} \right) {\hbox {d}}x \\&\quad \le \int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + \int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_{T}}\int \limits _\Omega {u_\varepsilon ^2} {\hbox {d}}x, \\&\quad \le \int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_T}, \end{aligned}$$

then by a direct integration, and by (A.7), we obtain (A.22).

By (A.21) and the \(L^p\) theory of linear parabolic equations, (A.23) is valid obviously. The proof is completed. \(\square \)

Lemma A.6

Assume \(N=2, 3\), \(m>1\), and (A.2) holds. Then, for any \(\varepsilon >0\), the problem (A.1) admits a unique global classical solution \((u_\varepsilon , v_\varepsilon , w_\varepsilon )\in C^{2+\alpha , 1+\alpha /2}(\Omega \times (0, +\infty ))\).

Proof

By Lemma A.1, suppose that \(T_{\max }<+\infty \). Then, take \(T=T_{\max }\) in Lemma A.5, recalling (A.4), (A.13), (A.21). It is a contradiction. We complete the proof. \(\square \)

In order to derive the global weak solution of (1.4), we also need some more estimates, that is

Lemma A.7

Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have

$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} \int \limits _\Omega {u_\varepsilon ^2} \mathrm{d}x + \int \limits _0^T {\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} \mathrm{d}x\mathrm{d}s \le {C_T},} \end{aligned}$$
(A.26)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,\infty )} {\left\| {\nabla {w_\varepsilon }} \right\| _{{L^\infty }}} \le {C_T}, \end{aligned}$$
(A.27)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} \int \limits _\Omega {{{\left| {\nabla \left( {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }} \right) } \right| }^2}} \mathrm{d}x + \int \limits _0^T {\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} \mathrm{d}x} \mathrm{d}t \le {{C_T}} , \end{aligned}$$
(A.28)

where \(C_T\) at most depends on \(\mu ,\chi ,\xi ,\beta ,{u_0},{v_0},{w_0},T\).

Proof

Multiplying the first equation of (A.1) by \({u_\varepsilon }\), and integrating it over \(\Omega \), since \({u_\varepsilon },{v_\varepsilon }\) are bounded, then by (A.14) and (A.21), we obtain

$$\begin{aligned} \frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^2} {\hbox {d}}x + \int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} {\text { }}&\le - \frac{1}{2}\chi \int \limits _\Omega {u_\varepsilon ^2} \Delta {v_\varepsilon }{\hbox {d}}x - \frac{1}{2}\xi \int \limits _\Omega {u_\varepsilon ^2} \Delta {w_\varepsilon }{\hbox {d}}x + {C_T}\\&\le {C_1}\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_2}\int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_T}\\&\le {C_1}\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_T}, \end{aligned}$$

then by a direct integration and (A.7), we obtain (A.26).

Add \(\nabla \) to the third equation of (A.1), by a direct calculation, we have

$$\begin{aligned} \nabla {w_\varepsilon }( \cdot ,t) = {e^{ - \int \limits _0^t {(2\eta {w_\varepsilon } + \delta {v_\varepsilon } - \eta ){\hbox {d}}s} }}\left( {w_{\varepsilon 0}} - \int \limits _0^t {\delta \nabla {v_\varepsilon }{w_\varepsilon } \cdot {e^{\int \limits _0^t {(2\eta {w_\varepsilon } + \delta {v_\varepsilon } - \eta ){\hbox {d}}s} }}{\hbox {d}}s}\right) \end{aligned}$$

since \({w_\varepsilon }\) and \({\left\| {\nabla {v_\varepsilon }} \right\| _{{L^\infty }}}\) are bounded, we obtain (A.27).

Multiplying the first equation of (A.1) by \(\frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}\), and integrating it over \(\Omega \), since \({u_\varepsilon },{v_\varepsilon }\) are bounded, then by (A.13) and (A.27), we have

$$\begin{aligned}&\frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}{\int \limits _\Omega {\left| {\nabla \left( {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }} \right) } \right| } ^2}{\hbox {d}}x + {\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| } ^2}{\hbox {d}}x\\&\quad \le - \chi \int \limits _\Omega {\nabla ({u_\varepsilon }\nabla {v_\varepsilon })} \frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}{\hbox {d}}x - \xi \int \limits _\Omega {\nabla ({u_\varepsilon }\nabla {w_\varepsilon })} \frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}{\hbox {d}}x\\&\qquad + \mu \int \limits _\Omega {{u_\varepsilon }\left( {1 + \frac{\beta }{\mu }{v_\varepsilon } - {u_\varepsilon }} \right) } \frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}{\hbox {d}}x\\&\quad \le {C_3}\int \limits _\Omega {{{\left| {\nabla ({u_\varepsilon }\nabla {v_\varepsilon })} \right| }^2}{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{\hbox {d}}x} + {C_4}\int \limits _\Omega {{{\left| {\nabla ({u_\varepsilon }\nabla {w_\varepsilon })} \right| }^2}{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{\hbox {d}}x}\\&\qquad + \frac{1}{2}\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} {\hbox {d}}x + {C_5}{\int \limits _\Omega {u_\varepsilon ^2\left( {1 + \frac{\beta }{\mu }{v_\varepsilon } - {u_\varepsilon }} \right) } ^2}{(u_\varepsilon ^2 + \varepsilon )^{\frac{{m - 1}}{2}}}{\hbox {d}}x\\&\quad \le {C_T}\left( {1 + \left( {\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2} + } {{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} \right) {\hbox {d}}x} \right) + \frac{1}{2}\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} {\hbox {d}}x, \end{aligned}$$

it means

$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{{\left| {\nabla \left( {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }} \right) } \right| }^2}} {\hbox {d}}x + \int \limits _\Omega {{{\left( {u_\varepsilon ^2 + \varepsilon } \right) }^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} {\hbox {d}}x \le {C_T}\left( {1 + \int \limits _\Omega {\left( {{{\left| {\Delta {v_\varepsilon }} \right| }^2} + {{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} \right) } {\hbox {d}}x} \right) , \end{aligned}$$

then by a direct integration and (A.7), (A.26), we obtain (A.28). \(\square \)

Proof of Lemma 3.1.

We take \(T={T_{\max }}\) and then the problem (A.1) admits a unique global classical solution \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon }) \in {C^{2 + \alpha ,1 + \alpha /2}}(\Omega \times (0,\infty ))\). Next, we will prove the existence of weak solutions. In what follows, we let ‘\(\rightarrow \)’ denote the strong convergence, and ‘\(\rightharpoonup \)’ denote the weak convergence.

By Lemma A.7, noticing that \(m>1\), we see that

$$\begin{aligned}&u_\varepsilon ^{\frac{{m + 1}}{2}} \rightharpoonup {u^{\frac{{m + 1}}{2}}}\quad in\;W_2^{1,1}({Q_T}),\\&{u_\varepsilon }\mathop \rightharpoonup \limits ^* u,\quad in\;{L^\infty }({Q_T}), \end{aligned}$$

then by Aubin-lions lemma [25], we have

$$\begin{aligned} u_\varepsilon ^{} \rightarrow {u^{}}\quad in\;{L^p}({Q_T})\;,\quad for\;any\;p \in (1,\infty ). \end{aligned}$$

Noticing that \({(a + b)^\alpha } \le {a^\alpha } + {b^\alpha }\) for any \(a,b>0,\alpha \in (0,1)\). When \(m\le 3\),

$$\begin{aligned} \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - {u^m}} \right| \le \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - u_\varepsilon ^m} \right| + \left| {u_\varepsilon ^m - {u^m}} \right| \le {\varepsilon ^{\frac{{m - 1}}{2}}}{u_\varepsilon } + \left| {u_\varepsilon ^m - {u^m}} \right| , \end{aligned}$$

when \(m>3\),

$$\begin{aligned} \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - {u^m}} \right|&\le \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - u_\varepsilon ^m} \right| + \left| {u_\varepsilon ^m - {u^m}} \right| \\&\le \frac{m-1}{2}\varepsilon (u_\varepsilon ^2 + \varepsilon )^{\frac{m-1}{2}-1} u_\varepsilon + \left| {u_\varepsilon ^m - {u^m}} \right| \\&\le c\varepsilon u_\varepsilon ^{m - 2} + c{\varepsilon ^{\frac{{m - 1}}{2}}}{u_\varepsilon } + \left| {u_\varepsilon ^m - {u^m}} \right| , \end{aligned}$$

which implies that

$$\begin{aligned} {(u_\varepsilon ^2 + \varepsilon )^{\frac{{m - 1}}{2}}}{u_\varepsilon } \rightarrow {u^m}\;in\;{L^p}({Q_T}),\;for\;any\;p \in (1, + \infty ). \end{aligned}$$

Noting that \(W_{p'}^{2,1}({Q_T}) \hookrightarrow {C^{\alpha ,\frac{\alpha }{2}}}({Q_T})\) for any \(0 < \alpha \le 2 - \frac{5}{{p'}} ({p'} > \frac{5}{2})\), so by (A.23), we have

$$\begin{aligned} {v_\varepsilon } \rightarrow v\quad uniformly. \end{aligned}$$

By (A.14), we have

$$\begin{aligned} \nabla {w_\varepsilon } \rightarrow \nabla w\quad in\;{L^p}({Q_T})\;, for\;any\;p \in (1, + \infty ), \end{aligned}$$

since \({w_\varepsilon }\) is bounded, and using (A.8) and Aubin-lions lemma, we also have

$$\begin{aligned} {w_\varepsilon } \rightarrow w\quad in\;{L^p}({Q_T})\;, for\;any\;p \in (1, + \infty ). \end{aligned}$$

Then, we have

$$\begin{aligned}&{u_\varepsilon }\nabla {v_\varepsilon } \rightharpoonup u\nabla v\quad in\;{L^q}({Q_T})\;for\;any\;1< q<\infty ,\\&\quad {u_\varepsilon } {v_\varepsilon } \rightarrow u v\quad in\;{L^q}({Q_T})\;for\;any\;1< q<\infty ,\\&\quad {u_\varepsilon }\nabla {w_\varepsilon } \rightarrow u\nabla w\quad in\;{L^q}({Q_T})\;, for\;any\;1< q< \infty ,\\&\quad {v_\varepsilon } {w_\varepsilon } \rightarrow v w\quad in\;{L^q}({Q_T})\;for\;any\;1< q< \infty ,\\&\quad w_\varepsilon ^2 \rightarrow {w^2}\quad in\;{L^q}({Q_T})\;for\;any\;1< q < \infty , \end{aligned}$$

recalling that for any \(\varphi ,\phi ,\psi \in {C^\infty }(\overline{{Q_T}} )\) with \({\left. {\frac{{\partial \varphi }}{{\partial n}}} \right| _{\partial \Omega }} = 0\),

$$\begin{aligned}&\int \limits _\Omega {{u_\varepsilon }(x,T)\varphi (x,T){\hbox {d}}x} - \int \limits _\Omega {{u_{\varepsilon 0}}\varphi (x,0){\hbox {d}}x} - \iint \limits _{{Q_T}} {{u_\varepsilon }{\varphi _t}}{\hbox {d}}x{\hbox {d}}t - \iint \limits _{{Q_T}} {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }\Delta \varphi {\hbox {d}}x{\hbox {d}}t}\\&\quad - \chi \iint \limits _{{Q_T}} {{u_\varepsilon }\nabla {v_\varepsilon }\nabla \varphi {\hbox {d}}x{\hbox {d}}t}- \xi \iint \limits _{{Q_T}} {{u_\varepsilon }\nabla {w_\varepsilon }\nabla \varphi {\hbox {d}}x{\hbox {d}}t} = \mu \iint \limits _{{Q_T}} {{u_\varepsilon }(1 - {u_\varepsilon })\varphi }{\hbox {d}}x{\hbox {d}}t + \beta \iint \limits _{{Q_T}} {{u_\varepsilon }{v_\varepsilon }\varphi }{\hbox {d}}x{\hbox {d}}t,\\&\quad \iint \limits _{{Q_T}} {{v_{\varepsilon t}}\phi {\hbox {d}}x{\hbox {d}}t + D\iint \limits _{{Q_T}} {\nabla {v_\varepsilon }\nabla \phi {\hbox {d}}x{\hbox {d}}t} + \iint \limits _{{Q_T}} {{u_\varepsilon }{v_\varepsilon }\phi {\hbox {d}}x{\hbox {d}}t} - \iint \limits _{{Q_T}} {{u_\varepsilon }\phi }{\hbox {d}}x{\hbox {d}}t = 0},\\&\quad \iint \limits _{{Q_T}} {{w_{\varepsilon t}}\psi {\hbox {d}}x{\hbox {d}}t + \delta \iint \limits _{{Q_T}} {{v_\varepsilon }{w_\varepsilon }\psi {\hbox {d}}x{\hbox {d}}t} - \eta \iint \limits _{{Q_T}} {\psi {w_\varepsilon }(1 - {w_\varepsilon }){\hbox {d}}x{\hbox {d}}t} = 0}. \end{aligned}$$

Let \(\varepsilon \rightarrow 0,\) we conclude that

$$\begin{aligned}&\int \limits _\Omega {u(x,T)\varphi (x,T){\hbox {d}}x} - \int \limits _\Omega {u(x,0)\varphi (x,0){\hbox {d}}x} - \iint \limits _{{Q_T}} {u{\varphi _t}}{\hbox {d}}x{\hbox {d}}t - \iint \limits _{{Q_T}} {{u^m}\Delta \varphi {\hbox {d}}x{\hbox {d}}t}\\&\quad - \chi \iint \limits _{{Q_T}} {u\nabla v\nabla \varphi {\hbox {d}}x{\hbox {d}}t}- \xi \iint \limits _{{Q_T}} {u\nabla w\nabla \varphi {\hbox {d}}x{\hbox {d}}t} = \mu \iint \limits _{{Q_T}} {u(1 - u)\varphi }{\hbox {d}}x{\hbox {d}}t + \beta \iint \limits _{{Q_T}} {uv\varphi }{\hbox {d}}x{\hbox {d}}t,\\&\quad \iint \limits _{{Q_T}} {{v_t}\phi {\hbox {d}}x{\hbox {d}}t + D\iint \limits _{{Q_T}} {\nabla v\nabla \phi {\hbox {d}}x{\hbox {d}}t} + \iint \limits _{{Q_T}} {uv\phi {\hbox {d}}x{\hbox {d}}t} - \iint \limits _{{Q_T}} {u\phi }{\hbox {d}}x{\hbox {d}}t = 0},\\&\quad \iint \limits _{{Q_T}} {{w_t}\psi {\hbox {d}}x{\hbox {d}}t + \delta \iint \limits _{{Q_T}} {vw\psi {\hbox {d}}x{\hbox {d}}t} - \eta \iint \limits _{{Q_T}} {\psi w(1 - w){\hbox {d}}x{\hbox {d}}t} = 0}. \end{aligned}$$

By (A.28), we have \(\nabla {u^m} \in {L^2}({Q_T})\). Then, we also have

$$\begin{aligned}&\int \limits _\Omega {u(x,T)\varphi (x,T){\hbox {d}}x} - \int \limits _\Omega {u(x,0)\varphi (x,0){\hbox {d}}x} - \iint \limits _{{Q_T}} {u{\varphi _t}}{\hbox {d}}x{\hbox {d}}t - \iint \limits _{{Q_T}} {\nabla {u^m}\nabla \varphi {\hbox {d}}x{\hbox {d}}t}\\&\quad - \chi \iint \limits _{{Q_T}} {u\nabla v\nabla \varphi {\hbox {d}}x{\hbox {d}}t}- \xi \iint \limits _{{Q_T}} {u\nabla w\nabla \varphi {\hbox {d}}x{\hbox {d}}t} = \mu \iint \limits _{{Q_T}} {u(1 - u)\varphi }{\hbox {d}}x{\hbox {d}}t + \beta \iint \limits _{{Q_T}} {uv\varphi }{\hbox {d}}x{\hbox {d}}t. \end{aligned}$$

Hence, (uvw) is a weak solution of (1.4), and all the estimations are hold by \(\varepsilon \rightarrow 0\).\(\square \)

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, D., Jin, C. & Xiang, Y. Stabilization to a cancer invasion model with remodeling mechanism and slow diffusion. Z. Angew. Math. Phys. 73, 201 (2022). https://doi.org/10.1007/s00033-022-01839-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01839-0

Keywords

Mathematics Subject Classification

Navigation