Well-Posedness of Difference Schemes for Semilinear Parabolic Equations with Weak Solutions

作者：Matus P P^*

来源：Computational Mathematics and Mathematical Physics, 2010, 50(12): 2044-2063.

DOI：10.1134/S0965542510120079

• weak solution
• initial-boundary value problem
• semilinear parabolic equation
• finite-difference scheme
• stability
• a priori estimates
• solution blow-up
• method of energy inequalities
• Chaplygin comparison theorem

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 <= t <= +infinity. 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.

• 出版日期2010-12