Volume 10, no. 2Pages 107 - 123

Local Solvability and Decay of the Solution of an Equation with Quadratic Noncoercive Nonlineatity

M.O. Korpusov, D.V. Lukyanenko, E.A. Ovsyannikov, A.A. Panin
An initial-boundary value problem for plasma ion-sound wave equation is considered. Boltzmann distribution is approximated by a quadratic function. The local (in time) solvability is proved and the analitycal-numerical investigation of the solution's decay is performed for the considered problem. The sufficient conditions for solution's decay and an upper bound of the decay moment are obtained by the test function method. In some numerical examples, the estimation is specified by Richardson's mesh refinement method. The time interval for numerical modelling is chosen according to the decay moment's analytical upper bound. In return, numerical calculations refine the moment and the space-time pattern of the decay. Thus, analytical and numerical parts of the investigation amplify each other.
Full text
blow-up; nonlinear initial-boundary value problem; Sobolev type equation; exponential nonlinearity; Richardson extrapolation.
1. Korpusov M.O. Critical Exponents of Instantaneous Blow-up or Local Solvability of Non-Linear Equations of Sobolev Type. Izvestiya: Mathematics, 2015, vol. 79, no. 5, pp. 955-1012. DOI: 10.1070/IM2015v079n05ABEH002768
2. Korpusov M.O., Lukyanenko D.V., Panin A.A., Yushkov E.V. Blow-up for One Sobolev Problem: Theoretical Approach and Numerical Analysis. Journal of Mathematical Analysis and Applications, 2016, vol. 442, no. 2, pp. 451-468. DOI: 10.1016/j.jmaa.2016.04.069
3. Korpusov M.O., Lukyanenko D.V., Panin A.A., Yushkov E.V. Blow-up Phenomena in the Model of a Space Charge Stratification in Semiconductors: Analytical and Numerical Analysis. Mathematical Methods in the Applied Sciences, 2017, vol. 40, no. 7, pp. 2336-2346. DOI: 10.1002/mma.4142
4. Korpusov M.O. The Finite-Time Blowup of the Solution of an Initial Boundary-Value Problem for the Nonlinear Equation of Ion Sound Waves. Theoretical and Mathematical Physics, 2016, vol. 187, no. 3, pp. 835-841. DOI: 10.1134/S0040577916060040
5. Panin A.A. On Local Solvability and Blow-up of the Solution of an Abstract Nonlinear Volterra Integral Equation. Mathematical Notes, 2015, vol. 97, no. 6, pp. 892-908. DOI: 10.1134/S0001434615050247
6. Hairer E. Solving of Ordinary Differential Equations. Berlin, Heidelberg, Spinger-Verlag, 2002.
7. Kalitkin N.N. [Numerical Methods for Solving Stiff Systems]. Matematicheskoe modelirovanie, 1995, vol. 7, no. 5, pp. 8-11. (in Russian)
8. Rosenbrock H.H. Some General Implicit Processes for the Numerical Solution of Differential Equations. The Computer Journal, 1963, vol. 5, issue 4, pp. 329-330. DOI: 10.1093/comjnl/5.4.329
9. Al'shin A.B., Al'shina E.A., Kalitkin N.N., Koryagina A.B. Rosenbrock Schemes with Complex Coefficients for Stiff and Differential-Algebraic Systems. Computational Mathematics and Mathematical Physics, 2006, vol. 46, no. 8, pp. 1320-1340. DOI: 10.1134/S0965542506080057
10. Al'shina E.A., Kalitkin N.N., Koryakin P.V. Diagnosis of Singularities of an Exact Solution in Computations with Accuracy Control. Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 10, pp. 1769-1779.
11. Al'shin A.B., Al'shina E.A. Numerical Diagnosis of Blow-up of Solutions of Pseudoparabolic Equations. Journal of Mathematical Sciences, 2008, vol. 148, no. 1, pp. 143-162. DOI: 10.1007/s10958-007-0542-2
12. Kalitkin N.N., Al'shin A.B., Al'shina E.A., Rogov B.V. Vychisleniya na kvaziravnomernykh setkakh [Calculations on Quasi-Uniform Grids]. Moscow, Fizmatlit, 2005.