Comparison theorem in blow-up problems for reaction diffusion equations and for their approximations

In this paper, blow-up sufficient conditions and upper bound of blow-up time for solution of Neumann and Dirichlet problems for reaction diffusion equations with non-linear gradient have been obtained. These equations have been found from the comparison of theorems, Jensen’s inequality and conservations laws. By using a similar proof approach for the finite-difference case, the finite-difference scheme was constructed, approximating the above-mentioned Neumann problem, for which sufficient conditions and upper bound of blow-up time, consistent with appropriate conditions and bound for the appropriate differential problem, have been obtained.