Global Existence and Uniform Blow-Up to a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising in a Thermal Explosion Theory

This paper deals with a nonlinear nonlocal parabolic system with nonlinear heat-loss boundary conditions, which arise in the thermal explosion model. Firstly, we prove a comparison principle for some kinds of parabolic systems under nonlinear boundary conditions. Using this, we improve a new theorem of the sub-and-super solution. Secondly, based on the new sub-and-super solution theorem, the sufficient conditions that the solution exists and blows up uniformly in finite time are presented. Then, we generalize some of the lemmas related to uniform blow-up solutions, which are used to introduce the uniform blow-up profiles of solutions. Finally, we give several numerical simulations to illustrate the existence and uniform blow-up of solutions.

Automated summary

This paper studies a nonlinear nonlocal parabolic system with nonlinear heat-loss boundary conditions, which are relevant to the thermal explosion model. The paper first proves a comparison principle for certain types of parabolic systems with nonlinear boundary conditions. Based on this, a new theorem for sub-and-super solutions is improved. The paper then presents sufficient conditions for the existence and blow-up of uniformly in finite time solutions based on the new theorem. Additionally, the paper generalizes some lemmas related to uniform blow-up solutions and provides numerical simulations to illustrate the existence and uniform blow-up of solutions.