标题： 具有逻辑源的完全抛物化学趋向性模型中肿瘤-免疫细胞相互作用 显示英文标题

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

摘要： 本研究探讨了一类具化学趋向性的系统中经典解的存在性问题，这些系统可表示为 \[\begin{cases} u_t = \Delta u-\chi \nabla\cdot(u \nabla v) + \mu_1 u^k -\mu_2 u^{k+1}, & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ v_t= \Delta v+\alpha w-\beta v-\gamma u v, & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ w_t= \Delta w-\delta u w+ \mu_3 w(1-w), & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial w}{\partial\nu}=0, & \text{on} \; \partial\Omega\times(0,T_{\text{max}}), \\ u(x,0)=u_0(x), \quad v(x,0)= v_0(x), \quad w(x,0)= w_0(x), & x\in\overline{\Omega}, \end{cases}\]，用于描述肿瘤（即 $w$）与免疫细胞（即 $u$）之间的相互作用，并带有逻辑斯蒂型源项 $\mu_1 u^k - \mu_2 u^{k+1}$， $k\geq1$，同时也考虑了化学信号（即 $v$）的存在。 模型参数$\chi, \mu_1,\mu_2, \mu_3, \alpha, \beta, \gamma$和$\delta$均为正值。 值$T_{\text{max}}$表示解定义的最大时间瞬间。 我们的重点是在 Neumann 边界条件下研究有界域$\Omega\subset \mathbb{R}^n, n \geq 3$中的全局存在性。 我们区分两种情形：$k>1$和$k=1$。 第一种情形允许在更小的假设下证明有界性，这些假设仅依赖于模型参数而非初始数据，而第二种情形则需要一个额外条件，将参数$\chi, \mu_2$、$n$与初始数据$\lVert v_0 \rVert_{L^\infty(\Omega)}$联系起来。 此模型可以看作是对[11]和[4]中先前研究模型的扩展，其中前者是一个只有两个方程的系统，而后者则是去除了逻辑斯蒂项的同一模型。 显示英文摘要

摘要： This work studies the existence of classical solutions to a class of chemotaxis systems reading \[\begin{cases} u_t = \Delta u-\chi \nabla\cdot(u \nabla v) + \mu_1 u^k -\mu_2 u^{k+1}, & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ v_t= \Delta v+\alpha w-\beta v-\gamma u v, & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ w_t= \Delta w-\delta u w+ \mu_3 w(1-w), & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial w}{\partial\nu}=0, & \text{on} \; \partial\Omega\times(0,T_{\text{max}}), \\ u(x,0)=u_0(x), \quad v(x,0)= v_0(x), \quad w(x,0)= w_0(x), & x\in\overline{\Omega}, \end{cases}\] 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 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], being the former a system with only two equations and the latter the same model without logistic.