Well-posedness of difference schemes for semilinear parabolic equations with weak solutions

P. P. Matus

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 12, pp. 2155-2175

Résumé

The well-posedness of difference schemes approximating initial-boundary value problem for parabolic equations with a nonlinear power-type source is studied. Simple sufficient conditions on the input data are obtained under which the weak solutions of the differential and difference problems are globally stable for all $0\leq t\leq+\infty$. It is shown that, if the condition fails, the solution can blow up (become infinite) in a finite time. A lower bound for the blow-up time is established. In all the cases, the method of energy inequalities is used as based on the application of the Chaplygin comparison theorem, Bihari-type inequalities, and their difference analogues. A numerical experiment is used to illustrate the theoretical results and verify two-sided blow-up time estimates.

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