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Nonlocal adhesion models for two cancer cell phenotypes in a multidimensional bounded domain

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Abstract

Cell–cell adhesion is an inherently nonlocal phenomenon. Numerous partial differential equation models with nonlocal term have been recently presented to describe this phenomenon, yet the mathematical properties of nonlocal adhesion model are not well understood. Here we consider a model with two kinds of nonlocal cell–cell adhesion, satisfying no-flux conditions in a multidimensional bounded domain. We show global-in-time well-posedness of the solution to this model and obtain the uniform boundedness of solution.

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Notes

  1. There is the other steady state, \(u^{*}=-\frac{\mu }{\lambda }v^{*}, v^{*}=\frac{k(1-\frac{m}{\lambda })}{1-\frac{\mu }{\lambda }}\), which are of different signs, hence unrealistic.

  2. For case II, we could change the shrinking rate and shape of E(x) as x approaches the boundary so that the regularity of adhesion terms is possibly worse.

  3. Eckardt et al. considered averaging nonlocal operators \(\mathcal A_r\) and \(\mathring{\nabla _r}\). Following notations in this paper, we have

    $$\begin{aligned} \mathcal A_Ru(x) = \frac{1}{R}\frac{1}{|B_R(0)|} \int \limits _{E(x)} u(x+y) \frac{y}{|y|} w(|y|) dy= \frac{1}{M_{11}R}\frac{1}{|B_R(0)|}\mathcal K [u,0] \end{aligned}$$

    by choosing \({\omega }(x)=\frac{y}{|y|} w(|y|)\) in (1.2).

  4. We follow a definition of \(C^1\)-functions on a submanifold \(\partial {\Omega }\times \mathbb {R}\) embedded in \(\mathbb {R}^n \times \mathbb {R}\). The open set E(V)is called a tubular neighborhood of \(\partial {\Omega }\).

  5. \(A_{\beta }\) is a \(W^{s,p}\)- realization of A if \( A= A_{\beta }\) in D(A) and the range of \(A_{\beta }\) is in \(W^{s,p}\).

References

  1. Alikakos, N.D.: \(L^{p}\) bounds of solutions of reaction–diffusion equations. Commun. Partial Differ. Equ. 4, 827–868 (1979). https://doi.org/10.1080/03605307908820113

    Article  MATH  Google Scholar 

  2. Amann, H.: Semigroups and nonlinear evolution equations. Linear Algebra Appl. 84, 3–32 (1986). https://doi.org/10.1016/0024-3795(86)90305-8

    Article  MathSciNet  MATH  Google Scholar 

  3. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H.J., Triebel, H. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, vol. 133, pp. 9–126 (1993). https://doi.org/10.1007/978-3-663-11336-2_1

  4. Anderson, A.R.A., Chaplain, M.A.J., Newman, E.L., Steele, R.J.C., Thompson, A.M.: Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129–154 (2000). https://doi.org/10.1080/10273660008833042

    Article  MATH  Google Scholar 

  5. Andasari, V., Chaplain, M.A.J.: Intracellular modelling of cell-matrix adhesion during cancer cell invasion. Math. Model. Nat. Phenomena 7, 29–48 (2012). https://doi.org/10.1051/mmnp/20127103

    Article  MathSciNet  MATH  Google Scholar 

  6. Armstrong, N.J., Painter, K.J., Sherratt, J.A.: A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243, 98–113 (2006). https://doi.org/10.1016/j.jtbi.2006.05.030

    Article  MathSciNet  MATH  Google Scholar 

  7. Armstrong, N.J., Painter, K.J., Sherratt, J.A.: Adding adhesion to a chemical signaling model for somite formation. Bull. Math. Biol. 71, 1–24 (2009). https://doi.org/10.1007/s11538-008-9350-1

    Article  MathSciNet  MATH  Google Scholar 

  8. Bellomo, N., Li, N.K., Maini, P.K.: On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18, 593–646 (2008). https://doi.org/10.1142/s0218202508002796

    Article  MathSciNet  MATH  Google Scholar 

  9. Bertozzi, A., Laurent, T., Rasado, J.: \(L^p\) theory for the multidimensional aggregation equation. Commun. Pure Appl. Math. 64, 45–83 (2011). https://doi.org/10.1002/cpa.20334

    Article  Google Scholar 

  10. Bertozzi, A., Laurent, T.: Finite time Blow-up of solutions of an Aggregation Equation in \(\mathbb{R}^n\). Commun. Math. Sci. 274, 717–735 (2007). https://doi.org/10.1007/s00220-007-0288-1

    Article  MATH  Google Scholar 

  11. Bitsouni, V., Chaplain, M.A.J., Eftimie, R.: Mathematical modelling of cancer invasion: the multiple roles of TGF-\(\beta \) pathway on tumour proliferation and cell adhesion. Math. Models Methods Appl. Sci. 27, 1929–1962 (2017). https://doi.org/10.1142/s021820251750035x

    Article  MathSciNet  MATH  Google Scholar 

  12. Bitsouni, V., Eftimie, R.: Non-local parabolic and hyperbolic models for cell polarisation in heterogeneous cancer cell populations. Bull. Math. Biol. 80, 2600–2632 (2018). https://doi.org/10.1007/s11538-018-0477-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Chaplain, M.A.J., Anderson, A.R.A.: Mathematical modelling of tissue invasion. In: Preziosi, L. (ed.) Cancer Modelling and Simulation, pp. 267–297. Chapman Hall/CRT, Cambridge (2003). https://doi.org/10.1201/9780203494899.ch10

    Chapter  Google Scholar 

  14. 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, 1685–1734 (2005). https://doi.org/10.1142/s0218202505000947

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, L., Painter, K.J., Surulescu, C., Zhigun, A.: Mathematical models for cell migration: a non-local perspective. Philos. Trans. R. Soc. B 375, 20190379 (2020). https://doi.org/10.1098/rstb.2019.0379

    Article  Google Scholar 

  16. Cieślak, T., Morales-Rodrigo, C.: Long-time behavior of an angiogenesis model with flux at the tumor boundary. Zeitschrift für angewandte Mathematik und Physik 64, 1625–1641 (2013). https://doi.org/10.1007/s00033-013-0302-8

    Article  MathSciNet  MATH  Google Scholar 

  17. Danchin, R.: A Lagrangian approach for the compressible Navier-Stokes equations. Annales de l’Institut Fourier 64(2), 753–791 (2014). https://doi.org/10.5802/aif.2865

    Article  MathSciNet  MATH  Google Scholar 

  18. Dai, F., Liu, B.: Global boundedness of classical solutions to a two species cancer invation haptotaxis model with tissue remodeling. J. Math. Anal. Appl. 483, 123583 (2020). https://doi.org/10.1016/j.jmaa.2019.123583

    Article  MathSciNet  MATH  Google Scholar 

  19. Delgado, M., Gayte, I., Morales-Rodrigo, C., Suárez, A.: An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary. Nonlinear Anal. Theory Methods Appl. 72, 330–347 (2010). https://doi.org/10.1016/j.na.2009.06.057

    Article  MathSciNet  MATH  Google Scholar 

  20. Delgado, M., Morales-Rodrigo, C., Suárez, A., Tello, J.I.: On a parabolic-elliptic chemotactic model with coupled boundary conditions. Nonlinear Anal. Real World Appl. 11, 3884–3902 (2010). https://doi.org/10.1016/j.nonrwa.2010.02.016

    Article  MathSciNet  MATH  Google Scholar 

  21. Denk, R., Hieber, M., Prüss, J.: Optimal \(L^{p}\)-\(L^{q}\)-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 193–224 (2007). https://doi.org/10.1007/s00209-007-0120-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Domschke, P., Trucu, D., Gerisch, A., Chaplain, M.A.J.: Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. J. Theor. Biol. 361, 41–60 (2014). https://doi.org/10.1016/j.jtbi.2014.07.010

    Article  MathSciNet  MATH  Google Scholar 

  23. Eckardt, M., Painter, K.J., Surulescu, C., Zhigun, A.: Nonlocal and local models for taxis in cell migration: a rigorous limit procedure. J. Math. Biol. 81, 1251–1298 (2020). https://doi.org/10.1007/s00285-020-01536-4

    Article  MathSciNet  MATH  Google Scholar 

  24. Evans, L.C.: Partial differential equations. Am. Math. Soc. (2010). https://doi.org/10.1090/gsm/019

    Article  MATH  Google Scholar 

  25. Franssen, L.C., Lorenzi, T., Burgess, A.E.F., Chaplain, M.A.J.: A mathematical framework for modelling the metastatic spread of cancer. Bull. Math. Biol. 81, 1965–2010 (2019)

    Article  MathSciNet  Google Scholar 

  26. Fetecau, R.C., Kovacic, M.: Swarm equilibria in domains with boundaries. SIAM J. Appl. Dyn. Syst. 16, 1260–1308 (2017). https://doi.org/10.1137/17m1123900

    Article  MathSciNet  MATH  Google Scholar 

  27. Fu, X.: Reaction–diffusion equations with nonlinear and nonlocal advection applied to cell co-culture. Ph.D. thesis, Universit/’e de Bordeaux (2019)

  28. Gerisch, A., Chaplain, M.A.J.: Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J. Theor. Biol. 250, 684–704 (2008). https://doi.org/10.1016/j.jtbi.2007.10.026

    Article  MathSciNet  MATH  Google Scholar 

  29. Gerisch, A., Painter, K.J.: Mathematical modeling of cell adhesion and its applications to developmental biology and cancer invasion. In: Chauviere, A., Preziosi, L., Verdier, C. (eds.) Cell Mechanics: From Single Scale-Based Models to Multiscale Modelling, pp. 319–350. CRC Press, Cambridge (2010). https://doi.org/10.1201/9781420094558-c12

    Chapter  MATH  Google Scholar 

  30. Giesselmann, J., Kolbe, N., Lukáčová-Medvid’ová, M., Sfakianakis, N.: Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model. Discrete Contin. Dyn. Syst. B 23, 4397–4431 (2018). https://doi.org/10.3934/dcdsb.2018169

    Article  MathSciNet  MATH  Google Scholar 

  31. Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, vol. 840 (1981). https://doi.org/10.1007/bfb0089647

  32. Hillen, T., Buttenschön, A.: Nonlocal adhesion models for microorganisms on bounded domains. SIAM J. Appl. Math. 80, 382–401 (2020). https://doi.org/10.1137/19m1250315

    Article  MathSciNet  MATH  Google Scholar 

  33. Hillen, T., Painter, K.J., Winkler, M.: Global solvability and explicit bounds for non-local adhesion models. Eur. J. Appl. Math. 29, 645–684 (2018). https://doi.org/10.1017/s0956792517000328

    Article  MathSciNet  MATH  Google Scholar 

  34. Lasiecka, I., Triggiani, R.: A cosine operator approach to modeling \(L_3(0, T; L_2(\Gamma ))\) Boundary input hyperbolic equations. Appl. Math. Optim. 7, 35–83 (1981). https://doi.org/10.1007/bf01442108

    Article  MathSciNet  MATH  Google Scholar 

  35. Ladyzhenskaya, O.A., Solonnikov, V.A., Uralćeva, N.N.: Linear and quasi-linear equations of parabolic type, vol. 23. American Mathematical Society, Providence (1988). https://doi.org/10.1090/mmono/023

    Book  Google Scholar 

  36. Lee, J.: Smooth manifolds. In: Introduction to Smooth Manifolds, pp. 1–31. Springer, New York (2013). https://doi.org/10.1007/978-1-4419-9982-5_1

  37. Litcanu, G., Morales-Rodrigo, C.: Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20, 1721–1758 (2010). https://doi.org/10.1142/s0218202510004775

    Article  MathSciNet  MATH  Google Scholar 

  38. Morales-Rodrigo, C.: Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours. Math. Comput. Modell. 47, 604–613 (2008). https://doi.org/10.1016/j.mcm.2007.02.031

    Article  MathSciNet  MATH  Google Scholar 

  39. Painter, K.J., Armstrong, N.J., Sherratt, J.A.: The impact of adhesion on cellular invasion processes in cancer and development. J. Theor. Biol. 264, 1057–1067 (2010). https://doi.org/10.1016/j.jtbi.2010.03.033

    Article  MathSciNet  MATH  Google Scholar 

  40. Painter, K.J., Bloomfield, J.M., Sherratt, J.A., Gerisch, A.: A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull. Math. Biol. 77, 1132–1165 (2015). https://doi.org/10.1007/s11538-015-0080-x

    Article  MathSciNet  MATH  Google Scholar 

  41. Perumpanani, A.J., Byrne, H.M.: Extracellular matrix concentration exerts selection pressure on invasive cells. Eur. J. Cancer 35, 1274–1280 (1999). https://doi.org/10.1016/s0959-8049(99)00125-2

    Article  Google Scholar 

  42. Sherratt, J.A., Gourley, S.A., Armstrong, N.J., Painter, K.J.: Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Eur. J. Appl. Math. 20, 123–144 (2009). https://doi.org/10.1017/s0956792508007742

    Article  MathSciNet  MATH  Google Scholar 

  43. Steinberg, M.: On the mechanism of tissue reconstruction by dissociated cells, I. Population kinetics, differential adhesiveness, and the absence of directed migration. Proc. Natl. Acad. Sci. 48, 1577–1582 (1962). https://doi.org/10.1073/pnas.48.9.1577

    Article  Google Scholar 

  44. Stinner, C., Surulescu, C., Winkler, M.: Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46, 1969–2007 (2014). https://doi.org/10.1137/13094058x

    Article  MathSciNet  MATH  Google Scholar 

  45. Tao, Y., Winkler, M.: A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source. SIAM J. Math. Anal. 43, 685–704 (2011). https://doi.org/10.1137/100802943

    Article  MathSciNet  MATH  Google Scholar 

  46. Tao, Y., Winkler, M.: A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy. Nonlinear Anal. 198, 111870 (2020). https://doi.org/10.1016/j.na.2020.111870

    Article  MathSciNet  MATH  Google Scholar 

  47. Tao, Y., Winkler, M.: Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. J. Differ. Equ. 268, 4973–4997 (2020). https://doi.org/10.1016/j.jde.2019.10.046

    Article  MathSciNet  MATH  Google Scholar 

  48. Tao, Y., Wang, M.: Global solution for a chemotactic-haptotactic model of cancer invasion. Nonlinearity 21, 2221–2238 (2008). https://doi.org/10.1088/0951-7715/21/10/002

    Article  MathSciNet  MATH  Google Scholar 

  49. Wu, L., Slepčev, D.: Nonlocal interaction equations in environments with heterogeneities and boundaries. Commun. Partial Differ. Equ. 40, 1241–1281 (2015). https://doi.org/10.1080/03605302.2015.1015033

    Article  MathSciNet  MATH  Google Scholar 

  50. Yeung, K.T., Yang, J.: Epithelial-mesenchymal transition in tumor metastasis. Mol. Oncol. 11, 28–39 (2017). https://doi.org/10.1002/1878-0261.12017

    Article  Google Scholar 

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Acknowledgements

Jaewook Ahn’s work is supported by NRF-2018R1D1A1B07047465. Jihoon Lee’s work is supported by SSTF-BA1701-05. Myeongju Chae’s work is supported by NRF-2018R1A1A3A04079376.

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Appendix

Appendix

Let \({\Omega }, {\mathcal {K}}, {\mathcal {S}}\) and \({\omega }\) be given as in Section 3. We define the operator \( F: W^{2,p}_B \times W^{2,p}_B \rightarrow L^2\times L^2\) by

$$\begin{aligned} F(u,v)&= (F_1(u,v), F_2(u,v)),\\ F_1(u,v)&= \Delta u-\nabla \cdot (u \,\mathcal {K}[u,v] )+(1-m)u +\displaystyle \frac{\lambda }{k} u(k-(u+v)) \\ F_2(u,v)&= \Delta v-\nabla \cdot (v \,\mathcal {S}[u,v] )+mu+v +\displaystyle \frac{\mu }{k} v(k-(u+v)). \end{aligned}$$

We denote the Gâteaux derivative of F at \(U=(u,v)\) by \(T_{U}\);

$$\begin{aligned} T_{U}(W) = \lim _{t\rightarrow 0} \frac{F(U+tW)- F(U)}{t} = (\delta _W F_1(U), \delta _W F_2(U)) \end{aligned}$$

where \( W= (w, z) \in W^{2,p}_B \times W^{2,p}_B\). By computation, we have

$$\begin{aligned} \delta _WF_1(0,k)&= \Delta w-mw \\ \delta _WF_2(0,k)&= \Delta z - \nabla \cdot ( k{\mathcal {S}}[w,0]+kS[w,z]+z{\mathcal {S}}[0,k]) + (m-\mu )w - \mu z\\ \delta _WF_1(0,0)&= \Delta w - mw\\ \delta _WF_2(0,0)&= \Delta z + mw + \mu z. \end{aligned}$$

We consider the two linearized equations at (0, k) and (0, 0), respectively, with initial data \((w_0, z_0)\in W^{2,p}_B \times W^{2,p}_B\);

$$\begin{aligned} \begin{aligned} \partial _t w&= \Delta w-mw\\ \partial _t z&= \Delta z - \nabla \cdot ( k{\mathcal {S}}[w,0]+kS[w,z]+z{\mathcal {S}}[0,k]) + (m-\mu )w - \mu z, \end{aligned} \end{aligned}$$
(3.21)

and

$$\begin{aligned} \begin{aligned} \partial _t w&= \Delta w-mw \\ \partial _t z&= \Delta z + mw + \mu z. \end{aligned} \end{aligned}$$
(3.22)

Equations (3.21), (3.22) are decoupled, and it is immediate that

$$\begin{aligned} \Vert w\Vert _{W^{1,p}({\Omega })} \le e^{-mt}\Vert w_0\Vert _{W^{1,p}}, \quad p\ge 1 \end{aligned}$$
(3.23)

from

$$\begin{aligned} \partial _t(e^{m t} w ) = \Delta (e^{m t} w ). \end{aligned}$$

Let \( \tilde{z}\) denote \(e^{\mu t} z\). Multiplying to the z- equation of (3.21) by \({e^{\mu t}}\), we have

$$\begin{aligned} \partial _t \tilde{z}- \Delta \tilde{z}= -\nabla \cdot ( 2k{\mathcal {S}}[e^{\mu t}w,0]+kS[0,\tilde{z}]+ \tilde{z}{\mathcal {S}}[0,k]) + (m-\mu ) e^{\mu t}w. \end{aligned}$$
(3.24)

It holds that

$$\begin{aligned} \frac{d}{dt}\int \limits _{{\Omega }}|\tilde{z}|\le |m-\mu | e^{\mu t} \int \limits |w| \le |m-\mu | e^{-(m-\mu )t} \Vert w_0\Vert _{L^1({\Omega })}, \end{aligned}$$

which implies

$$\begin{aligned} \int \limits _{{\Omega }}|\tilde{z}|&\le \int \limits _{{\Omega }}|z_0| - \frac{|m-\mu |}{m-\mu }(e^{-(m-\mu )t}-1)\int \limits _{{\Omega }}|w_0| \end{aligned}$$

and

$$\begin{aligned} \int \limits _{{\Omega }}|z |&\le e^{-\mu t} \int \limits _{{\Omega }}|z_0| + (e^{-mt}- e^{-\mu t})\int \limits _{{\Omega }}|w_0| , \end{aligned}$$
(3.25)

where we abuse the notation by \( | m-\mu |/(m-\mu ) =0\) if \( \mu =m\). When \(m > \mu \), it holds that

$$\begin{aligned} \int \limits | \tilde{z}| \le \Vert z_0\Vert _{L^1({\Omega })} + \Vert w_0\Vert _{L^1({\Omega })}. \end{aligned}$$
(3.26)

In what follows, we find that different signs of \({\mp } \mu z\) in (3.21) and (3.22) imply that (0, k) is linearly stable and (0, 0) is linearly unstable as expected.

Proposition 1

Each of the linearized equations (3.21), (3.22) have a unique global solution (wz) for each in

$$\begin{aligned} C([0,t); W^{1,p}(\Omega ))\cap W^{1,p}(0,t; L^{p}(\Omega ))\cap L^{p}(0,t;W^{2,p}(\Omega )) \end{aligned}$$

for any \(t>0\). When \(m> \mu \), the solution (wz) for (3.21) is asymptotically stable such that

$$\begin{aligned} \Vert z\Vert _{L^p({\Omega }) }\le e^{-\mu t}\Vert z_0\Vert _{L^p({\Omega })} \quad \hbox { for } p\ge 1. \end{aligned}$$
(3.27)

The solution (wz) for (3.22) grows exponentially in its \(L^1\)-norm if the initial data is non-negative;

$$\begin{aligned} \int \limits _{\Omega }|z| \ge e^{\mu t}\int \limits _{\Omega }|z_0|. \end{aligned}$$
(3.28)

Proof

Due to the a priori estimates (3.23) and (3.25), the global well-posedness part for (3.21) follows from the same argument in Sect. 2.3 or Sect. 3.2. Repeating the argument of Lemma 6 to (3.21) it holds that

$$\begin{aligned} \Vert z\Vert _{L^{\infty }({\Omega })}\le C(\Vert w_0\Vert _{L^1\cap L^{\infty }({\Omega })}, \Vert z_0\Vert _{L^1\cap L^{\infty }({\Omega })}). \end{aligned}$$
(3.29)

For details, see (3.31)–(3.33) for \(\tilde{z}\), where the similar estimates are given. By Lemma 9, it also holds that

$$\begin{aligned} \Vert z\Vert _{W^{1,p}} \le C(\Vert w_0\Vert _{L^1\cap L^{\infty }({\Omega })}, \Vert z_0\Vert _{L^1\cap L^{\infty }({\Omega })} ) \end{aligned}$$
(3.30)

for any \(p\ge 1\). Let us prove (3.28) first. The solution (wz) remains non-negative and we have

$$\begin{aligned} \int \limits _{{\Omega }}w&= e^{-mt}\int \limits _{{\Omega }}w_0,\\ \frac{d}{dt}(e^{-\mu t} \int \limits _{{\Omega }}z)&= m e^{-(\mu + m)t} \int \limits _{{\Omega }}w_0. \end{aligned}$$

Integrating the second equation, we have (3.28).

For (3.27), we proceed as in Lemma 6. Multiplying \(|\tilde{z}|^{p-2}\tilde{z}\) into (3.24) for \(p \ge 2\), we have

$$\begin{aligned} \frac{1}{p} \frac{d}{dt}\int \limits _{{\Omega }}|\tilde{z}|^{p}+ \frac{4(p-1)}{p^2}\int \limits _{{\Omega }}|\nabla \tilde{z}^{\frac{p}{2}}|^2 =&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot ( 2k{\mathcal {S}}[e^{\mu t}w,0]+kS[0,\tilde{z}]+ \tilde{z}{\mathcal {S}}[0,k]) \end{aligned}$$
(3.31)
$$\begin{aligned}&+ (m-\mu ) \int \limits _{{\Omega }}e^{\mu t}w |\tilde{z}|^{p-2}\tilde{z}. \end{aligned}$$
(3.32)

By Lemma 7, (3.23), (3.30) and using \(m >\mu \), we have

$$\begin{aligned}&\Vert \nabla \cdot {\mathcal {S}}[ e^{\mu t}w, 0]\Vert _{L^{\infty }({\Omega })}\le C\Vert e^{\mu t}w\Vert _{W^{1,q}({\Omega })} \le C \Vert w_0\Vert _{W^{1,q}({\Omega })} (q >n)\\&\Vert {\mathcal {S}}[ 0, \tilde{z}] \Vert _{L^{\infty }({\Omega })}\le C\Vert \tilde{z}\Vert _{L^1({\Omega })} \le C(\Vert w_0\Vert _{L^1}, \Vert z_0\Vert _{L^1})\\&\Vert \nabla {\mathcal {S}}[0, k]\Vert _{L^{\infty }({\Omega })}\le C \end{aligned}$$

and estimate the right hand side of (3.31) as follows

$$\begin{aligned}&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot ( 2k{\mathcal {S}}[e^{\mu t}w,0]) + (m-\mu ) \int \limits _{{\Omega }}e^{\mu t}w |\tilde{z}|^{p-2}\tilde{z}\le C\int \limits _{{\Omega }}|\tilde{z}|^{p-1}\\&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot kS[0,\tilde{z}] \le \frac{p-1}{p^2} \int \limits _{{\Omega }}| \nabla \tilde{z}^{\frac{p}{2}|}|^2 + C(p-1) \int \limits \tilde{z}^{p-2} \Vert {\mathcal {S}}[0, \tilde{z}\Vert _{{L^{\infty }({\Omega })}}^2 \\&\le \frac{p-1}{p^2} \int \limits _{{\Omega }}| \nabla \tilde{z}^{\frac{p}{2}}|^2 + C \frac{p-1}{p} \left( |{\Omega }| +(p-2)\int \limits _{{\Omega }}|\tilde{z}|^{p}\right) ,\\&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot ( \tilde{z}{\mathcal {S}}[0,k]) \le \frac{1}{p^2} \int \limits _{{\Omega }}| \nabla \tilde{z}^{\frac{p}{2}}|^2 + C\int \limits _{{\Omega }}|\tilde{z}|^{p}\Vert \nabla {\mathcal {S}}[0, k]\Vert _{{L^{\infty }({\Omega })}}. \end{aligned}$$

Summing up, we have

$$\begin{aligned} \frac{1}{p} \frac{d}{dt}\int \limits _{{\Omega }}|\tilde{z}|^{p}+ \frac{3(p-1)}{p^2}\int \limits _{{\Omega }}|\nabla \tilde{z}^{\frac{p}{2}}|^2 \le C+ C\int \limits _{{\Omega }}|\tilde{z}|^{p}, \quad p\ge 2, \end{aligned}$$

where C is a uniform constant depending on \(\Vert w_0\Vert _{L^1({\Omega })}\), \(\Vert z_0\Vert _{L^1({\Omega })}\), and given constants \(\mu , m, k\) etc.. As was derived from (2.30) for v in Lemma 6, it holds that

$$\begin{aligned} \sup _ {0<t \le T} \Vert \tilde{z}\Vert _{L^{p_k}({\Omega })} \le C(\Vert z_0\Vert _{L^1({\Omega })}, \Vert z_0\Vert _{{L^{\infty }({\Omega })}}) \sup _ {0 <t \le T}\left( \Vert \tilde{z}\Vert _{L^1({\Omega })} +C \right) \quad p_k= 2^k, k=0, 1\dots . \end{aligned}$$

and

$$\begin{aligned} \sup _ {0 <t \le T} \Vert \tilde{z}\Vert _{L^{\infty }({\Omega })}\le C(\Vert w_0\Vert _{L^1\cap L^{\infty }({\Omega })}, \Vert z_0\Vert _{L^1\cap L^{\infty }({\Omega })}). \end{aligned}$$
(3.33)

That implies (3.27). \(\square \)

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Ahn, J., Chae, M. & Lee, J. Nonlocal adhesion models for two cancer cell phenotypes in a multidimensional bounded domain. Z. Angew. Math. Phys. 72, 48 (2021). https://doi.org/10.1007/s00033-021-01485-y

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