基于 logistic 来源的全抛物性化学趋导模型中的肿瘤-免疫细胞相互作用
偏微分方程分析
2025-04-15 v1
摘要
本文研究了关于以下类化学趋导系统的经典解的存在性：\n\n { \n u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + μ 1 u k − μ 2 u k + 1 , in Ω × ( 0 , T max ) , \n v t = Δ v + α w − β v − γ u v , in Ω × ( 0 , T max ) , \n w t = Δ w − δ u w + μ 3 w ( 1 − w ) , in Ω × ( 0 , T max ) , \n ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , on ∂ Ω × ( 0 , T max ) , \n u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ Ω ‾ , \n \n \n\begin{cases}\n 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}}), \\\n v_t= \Delta v+\alpha w-\beta v-\gamma u v, & \text{in} \; \Omega\times(0,T_{\text{max}}), \\\n w_t= \Delta w-\delta u w+ \mu_3 w(1-w), & \text{in} \; \Omega\times(0,T_{\text{max}}), \\\n \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}}), \\\n u(x,0)=u_0(x), \quad v(x,0)= v_0(x), \quad w(x,0)= w_0(x), & x\in\overline{\Omega},\n\end{cases}\n \n ⎩ ⎨ ⎧ \n u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + μ 1 u k − μ 2 u k + 1 , \n v t = Δ v + α w − β v − γ uv , \n w t = Δ w − δ u w + μ 3 w ( 1 − w ) , \n ∂ ν ∂ u = ∂ ν ∂ v = ∂ ν ∂ w = 0 , \n u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , in Ω × ( 0 , T max ) , in Ω × ( 0 , T max ) , in Ω × ( 0 , T max ) , on ∂ Ω × ( 0 , T max ) , x ∈ Ω , \n \n w w w u u u μ 1 u k − μ 2 u k + 1 \mu_1 u^k - \mu_2 u^{k+1} μ 1 u k − μ 2 u k + 1 k ≥ 1 k\geq1 k ≥ 1 v v v χ , μ 1 , μ 2 , μ 3 , α , β , γ \chi, \mu_1,\mu_2, \mu_3, \alpha, \beta, \gamma χ , μ 1 , μ 2 , μ 3 , α , β , γ δ \delta δ T max T_{\text{max}} T max Ω ⊂ R n , n ≥ 3 \Omega\subset \mathbb{R}^n, n \geq 3 Ω ⊂ R n , n ≥ 3 k > 1 k>1 k > 1 k = 1 k=1 k = 1 χ , μ 2 \chi, \mu_2 χ , μ 2 n n n ∥ v 0 ∥ L ∞ ( Ω ) \lVert v_0 \rVert_{L^\infty(\Omega)} ∥ v 0 ∥ L ∞ ( Ω )
引用
@article{arxiv.2504.10376,
title = {Tumor-immune cell interactions by a fully parabolic chemotaxis model with logistic source},
author = {Rafael Díaz Fuentes},
journal= {arXiv preprint arXiv:2504.10376},
year = {2025}
}