具有非局部项的免疫反应扩散系统解的长期存在性——一种面向解释的证明
Longterm existence of solutions of a reaction diffusion system with non-local terms modeling an immune response -- an interpretation-orientated proof
AI总结：
本文针对模拟肝脏感染且含非局部项的反应扩散系统，提出一种面向解释的证明方法，通过展示解在 $L^1$、$L^2$ 和 $L^\infty$ 范数下的有界性，确立了其全局存在性，并揭示了强烈免疫反应引发严重炎症的实际意义。
摘要：
本文展示了模拟肝脏感染的反应扩散系统解的全局存在性和有界性。存在性证明逐步呈现，重点在于在所模拟的肝脏感染背景下对中间结果进行解释。病毒与免疫系统细胞之间源于淋巴结免疫反应的动力学非局部效应，导出了一个积分偏微分方程。虽然抛物型偏微分方程的存在性定理是该领域的教科书示例，但额外的积分项需要新的方法来证明解的全局存在性。这允许建立一个侧重于解释的存在性证明，从而同时增进对系统和建模视角的洞察。我们展示了在 $L^1(Ω)$- 和 $L^2(Ω)$- 范数下解的有界性，并利用这些结果证明了解的全局存在性和有界性。证明的核心要素是处理反应项中起相反作用的机制，这些机制出现在所有种群动力学模型中，并导致具有相反单调性行为的反应项。在模拟肝脏感染的背景下，$L^\infty(Ω)$- 范数下的有界性具有实际意义：强烈的免疫反应会导致肝组织严重发炎。严重的炎症会对感染者的健康产生负面影响，并导致严重的继发疾病。所得的粗略估计值与数值测试进行了比较。
英文摘要：
This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time. We show the boundedness of the solution in the $L^1(Ω)$- and the $L^2(Ω)$-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the $L^\infty(Ω)$-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases. The gained rough estimates are compared with numerical tests.
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