Stability of Finite-difference Schemes for Semilinear Multidimensional Parabolic Equations

Abstract

Abstract Abstract — We have studied the stability of finite-difference schemes approximating initial-boundary value problem (IBVP) for multidimensional parabolic equations with a nonlinear source of a power type. We have obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all t. It is shown that if these conditions are not satisfied, then the solution can blow-up (go to infinity) in finite time. The lower bound of the blow-up time has been determined. The stability of the difference solution has been proven. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear ODEs, Bihari-type inequalities and their discrete analogs.