Global existence for a class of non-equilibrium reaction-diffusion systems with flux limitation

For a class of general reaction-diffusion systems with flux limitation in one dimension, when homogeneous Neumann boundary conditions are considered, we will prove global upper and lower boundness estimates for the positive classical solutions and boundness for their first-order spatial derivatives. In particular, the non-equilibrium radiation flux-limited diffusion coupled with material heat conduction model is included as a special case of the general reaction-diffusion systems. The key point of our method is to at first prove that the second-order spatial derivative of the solution of the second diffusion equation can be linearly controlled by the first-order spatial derivative of the solution of the flux-limited diffusion equation. Then, with this control, we can apply the Bernstein method to the flux-limited diffusion equation, while a stratifying time technique is introduced to get a complete first order spatial derivative estimate. With these global prior estimates, the global existence and uniqueness of positive classical solutions for the general flux-limited diffusion systems are proved.& COPY; 2023 Elsevier Ltd. All rights reserved.