A uniform bound for solutions to a thermo-diffusive system

We obtain uniform in time bounds for the solutions to a class of thermo-diffusive systems with classical and fractional diffusions. In the classical diffusion case, the nonlinearities are assumed to be at most exponentially growing, while for the fractional diffusions they should grow slightly sub-exponentially. The proofs are based on the strategy of Herrero, Lacey, and Velazquez.