Tumor-immune cell interactions by a fully parabolic chemotaxis model with logistic source

This work studies the existence of classical solutions to a class of chemotaxis systems by$ \begin{align} \begin{cases} u_t = \Delta u-\chi \nabla\cdot(u \nabla v) + \mu_1 u^k -\mu_2 u^{k+1}, & \text{in} \; \Omega\times(0, T_{\text{max}}), \\ v_t = \Delta v+\alpha w-\beta v-\gamma u v, & \text{in} \; \Omega\times(0, T_{\text{max}}), \ w_t = \Delta w-\delta u w+ \mu_3 w(1-w), & \text{in} \; \Omega\times(0, T_{\text{max}}), \\ \frac{{\partial u}}{{\partial \nu }} = \frac{{\partial v}}{{\partial \nu }} = \frac{{\partial w}}{{\partial \nu }} = 0, & \text{on} \; \partial\Omega\times(0, T_{\text{max}}), \\ u(x, 0) = u_0(x), \quad v(x, 0) = v_0(x), \quad w(x, 0) = w_0(x), & x\in\overline{\Omega}, \end{cases} \end{align} $that model interactions between tumor (i.e., $ w $) and immune cells (i.e., $ u $) with a logistic-type source term $ \mu_1 u^k - \mu_2 u^{k+1} $, $ k\geq1 $, also in presence of a chemical signal (i.e., $ v $). The model parameters $ \chi, \mu_1, \mu_2, \mu_3, \alpha, \beta, \gamma $, and $ \delta $ are all positive. The vector $ \nu $ is the outward unit normal on $ \partial \Omega $. The value $ T_{\text{max}} $ indicates the maximum instant of time up to which solutions are defined. Our focus is on examining the global existence in a bounded domain $ \Omega\subset \mathbb{R}^n, n \geq 3 $, under Neumann boundary conditions. We distinguish between two scenarios: $ k>1 $ and $ k = 1 $. The first case allows to prove boundedness under smaller assumptions relying only on the model parameters instead of on the initial data, while the second case requires an extra condition relating the parameters $ \chi, \mu_2 $, $ n $, and the initial data $ \lVert v_0 \rVert_{L^\infty(\Omega)} $. This model can be seen as an extension of those previously examined in [11] and [4], the former being a system with only two equations and the latter the same model without logistic.