Blow-Up Problem in Nonlinear Parabolic Equations

The aim of the paper is to present an introduction to the subject of the formation of singularity in nonlinear evolution problems, usually called an explosion. In short, we are interested in a situation where, starting from a smooth initial configuration and after the first period of classical evolution, the solution (or in some cases its derivatives) becomes infinite in a finite time due to the cumulative effect of nonlinearities. We concentrate on problems related to differential equations of parabolic type, or on systems of such equations. Two new methods for the numerical integration of Cauchy problems for ODEs with inflatable solutions described. The first method based on the application of a differential transformation, where the first derivative (defined in the original equation) is selected as a new independent variable. The second method is based on the introduction of a new nonlocal variable that reduces ODE to a system of related ODEs. Both methods lead to problems whose solutions do not have explosive singular points; therefore, standard numerical methods can be applied. The effectiveness of the proposed methods is illustrated by several test tasks.