Well-Posedness and Blow Up for IBVP for Semilinear Parabolic Equations and Numerical Methods

Abstract

Abstract We have studied the stability of finite-difference schemes approximating boundary value problems for parabolic equations with a nonlinear and nonmonotonic source of the power type. We have obtained simple sufficient input data conditions, in which the solution of the differential problem is globally stable for all 0 ≤ t ≤ +∞. It is shown that if these conditions fail, 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 solution of BVP for the nonlinear convection-diffusion equation has been investigated. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear differential equations, Bihari-type inequalities and their discrete analogs.