Blow-up for one Sobolev problem: Theoretical approach and numerical analysis

Abstract

This work develops the theory of the blow-up phenomena for one Sobolev initial–boundary value problem that arises in the theory of ion sound waves. This problem is considered with classical Neumann and nonclassical nonlocal boundary conditions. In both cases global unsolvability by using the method of test functions and local solvability by using the contracting mapping method are proved. The estimate of solvability time for classical solution is obtained. These estimations are used in numerical algorithm which allows us to specify the process of the solution's blow-up by using Richardson extrapolation. Some numerical experiments are presented in order to demonstrate the effectiveness of the proposed methods. The MATLAB code that realizes these numerical experiments is available.