On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations

Abstract

For a broad class of initial-boundary value problems for quasilinear parabolic equations with nonlinear source and their approximations, we show that if the initial energy is negative, then the solution always blows up in finite time. This is especially important for finding sufficiently simple and easy-to-verify conditions guaranteeing the presence of physical effects such as heat localization in peaking modes or thermal explosions and for deriving two-sided estimates of the solution lifespan. We construct the corresponding new classes of difference schemes for which grid analogs of integral conservation laws hold. We show that, to obtain efficient two-sided estimates for the blow-up time of the solution of the differential problem, in practice, one should use difference schemes with explicit as well as implicit approximation to the source.