Conservation laws and their significance in blow-upin nonlinear problems for parabolic equations

In this work for the Dirichlet problem for a parabolic equation with a nonlinear source term, the conditions on a source function, a coefficient of the equation and initial data under which the solution of the problem blows up in finite time were found and blow-up time was estimated. In the prove of blowing up of solutions and obtaining the estimate of blow up time, integral conservation laws are important. In the paper it shows that the obtained result can be used to explore models of various physical processes. Using a special Steklov technique of averaging of nonlinear coefficients, the difference schemes which satisfies the grid analogue of integral conservation laws were constructed.