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This paper shows the global existence and boundedness of solutions of areaction diffusion system modeling liver infections. The existence proof ispresented step by step and the focus lies on the interpretation of intermediateresults in the context of liver infections which is modeled. Non-local effectsin the dynamics between the virus and the immune system cells coming from theimmune response in the lymphs lead to an integro-partial differential equation.While existence theorems for parabolic partial differential equations aretextbook examples in the field, the additional integral term requires newapproaches to proving the global existence of a solution. This allows to set upan existence proof with a focus on interpretation leading to more insight inthe system and in the modeling perspective at the same time.  We show the boundedness of the solution in the $L^1(\Omega)$- and the$L^2(\Omega)$-norms, and use these results to prove the global existence andboundedness of the solution. A core element of the proof is the handling ofoppositely acting mechanisms in the reaction term, which occur in allpopulation dynamics models and which results in reaction terms with oppositemonotonicity behavior. In the context of modeling liver infections, theboundedness in the $L^\infty(\Omega)$-norm has practical relevance: Largeimmune responses lead to strong inflammations of the liver tissue. Stronginflammations negatively impact the health of an infected person and lead tograve secondary diseases. The gained rough estimates are compared withnumerical tests.

partial differential equations in applied mathematics

Longterm existence of solutions of a reaction diffusion system with  non-local terms modeling an immune response -- an interpretation-orientated  proof