Maximal regularity and optimal control for a non-local Cahn-Hilliard tumour growth model (Q6616064)

This article studies a nonlocal Cahn-Hilliard (CH) system motivated by tumor growth models with the aim of establishing maximal regularity and optimal control. The CH system is posed on a bounded domain \(\Omega\subset\mathbb{R}^N\), with \(N=2,3\), and with boundary of class \(\mathcal{C}^2\). The author takes constant mobility, \(m\equiv1\), and a regular potential defined on \(\mathbb{R}\), e.g. \(F(r)=\frac{1}{4}\left( 1-r^2 \right)^2\). The nonlocal term appears in the chemical potential as\N\[\N\mu=F'(\varphi)+(J*1)\varphi-J*\varphi-\chi\sigma,\N\]\Nwhere \(\varphi\) is the order-parameter, \(F\) is the regular potential, \(J\) is a sufficiently regular symmetric convolution kernel, \(\sigma\) is the nutrient concentration, and \(\chi\ge0\) represents the intensity of the chemotaxis cross-diffusion effect. As in the case of the usual CH equations, the system examined here is an \(H^{-1}\)-gradient flow of an associated energy functional.\N\NThe main result on the existence and uniqueness of local maximal solutions requires, among other assumptions, that\N\begin{itemize}\N\item \(J\in W^{1,1}_{\mathrm{loc}}\left( \mathbb{R}^N \right)\),\N\item \(J\) is radially symmetric and nondecreasing (that is, \(J(\cdot)=\tilde{J}(|\cdot|)\) for nondecreasing \(\tilde{J}:\mathbb{R}_+\rightarrow\mathbb{R}\)),\N\item there exists \(R_0\) such that \(r\mapsto J''(r)\) and \(r\mapsto \tilde{J}'(r)/r\) are monotone on \((0,R_0)\),\N\item there exists \(C_N>0\) such that \(\left| D^3 J(x) \right| \le C_N|x|^{-N-1}\) for any \(x\in\mathbb{R}^3\setminus\{0\}\),\N\end{itemize}\Nor\N\begin{itemize}\N\item \(J\in W^{2,1}_{\mathrm{loc}}\left( \mathbb{R}^N \right)\).\N\end{itemize}