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
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.
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.
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).
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 \(\partial {\Omega }\).
\(A_{\beta }\) is a \(W^{s,p}\)- realization of A if \( A= A_{\beta }\) in D(A) and the range of \(A_{\beta }\) is in \(W^{s,p}\).
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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
We denote the Gâteaux derivative of F at \(U=(u,v)\) by \(T_{U}\);
where \( W= (w, z) \in W^{2,p}_B \times W^{2,p}_B\). By computation, we have
We consider the two linearized equations at (0, k) and (0, 0), respectively, with initial data \((w_0, z_0)\in W^{2,p}_B \times W^{2,p}_B\);
and
Equations (3.21), (3.22) are decoupled, and it is immediate that
from
Let \( \tilde{z}\) denote \(e^{\mu t} z\). Multiplying to the z- equation of (3.21) by \({e^{\mu t}}\), we have
It holds that
which implies
and
where we abuse the notation by \( | m-\mu |/(m-\mu ) =0\) if \( \mu =m\). When \(m > \mu \), it holds that
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 (w, z) for each in
for any \(t>0\). When \(m> \mu \), the solution (w, z) for (3.21) is asymptotically stable such that
The solution (w, z) for (3.22) grows exponentially in its \(L^1\)-norm if the initial data is non-negative;
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
For details, see (3.31)–(3.33) for \(\tilde{z}\), where the similar estimates are given. By Lemma 9, it also holds that
for any \(p\ge 1\). Let us prove (3.28) first. The solution (w, z) remains non-negative and we have
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
By Lemma 7, (3.23), (3.30) and using \(m >\mu \), we have
and estimate the right hand side of (3.31) as follows
Summing up, we have
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
and
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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DOI: https://doi.org/10.1007/s00033-021-01485-y