# Existence of Global Classical and Weak Solutions to a Prion Equation with Polymer Joining

## TL;DR

This paper proves the global existence and uniqueness of classical solutions for a nonlinear prion proliferation equation with bounded reaction rates, and the existence of weak solutions for unbounded rates, using advanced mathematical methods.

## Contribution

It establishes the first rigorous proof of global solutions for a complex prion model involving polymer joining, splitting, and polymerization.

## Key findings

- Global existence and uniqueness of classical solutions for bounded reaction rates.
- Existence of weak solutions for unbounded reaction rates.
- Application of evolution operator theory and compactness arguments.

## Abstract

We consider a nonlinear integro-differential equation for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The equation can be written as a quasilinear Cauchy problem. For bounded reaction rates we prove global existence and uniqueness of classical solutions by means of evolution operator theory. We also prove global existence of weak solutions for unbounded reaction rates by a compactness argument.