Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor (Q6602161)

Let \(\Omega\) be a bounded domain of \(\mathbb{R}^n\), \(n\ge 1\), with smooth boundary. A chemotaxis model describing the interactions between tumor cells with density \(z\) and cytotoxic T-lymphocytes with density \(u\), through a chemoattractant with concentration \(v\) and complexes formed by the lymphocytes and the tumor cells with density \(w\), is proposed in [\textit{M. AI-Tameemi} et al., Biol. Direct 7, Paper No. 31, 22 p. (2012; \url{doi:10.1186/1745-6150-7-31})] and reads\N\begin{align*}\N\partial_t u & = \operatorname{div}(\nabla u - u \nabla v) \;\; & \text{ in }\;\; (0,\infty)\times\Omega, \\\N\partial_t v & = \Delta v - v + w \;\; & \text{ in }\;\; (0,\infty)\times\Omega, \\\N\partial_t w & = d \Delta w - w + auz \;\; & \text{ in }\;\; (0,\infty)\times\Omega, \\\N\partial_t z & = \Delta z - uz + w \;\; & \text{ in }\;\; (0,\infty)\times\Omega,\N\end{align*}\Nsupplemented with homogeneous Neumann boundary conditions and non-negative initial conditions, the parameters \(d\) and \(a\) being positive. Since this system features an indirect production mechanism of the chemoattractant, it is expected that it may delay or even prevent the finite time blow-up observed in the classical parabolic-parabolic Keller-Segel chemotaxis system in two space dimensions or higher. It is indeed the case in space dimension \(n\in\{1,2,3\}\), for which the existence and uniqueness of a classical solution are established. An important step in the proof is the derivation of an \(L^2\)-estimate on \(w\) which is obtained by a duality argument performed on the equation solved by \(w+az\). \(L^\infty\)-boundedness of the solution is also shown when \(a\le 1\).