Local Solvability and Decay of the Solution of an Equation with Quadratic Noncoercive Nonlineatity
DOI:
https://doi.org/10.14529//mmp170209Keywords:
, blow-up, nonlinear initial-boundary value problem, Sobolev type equation, exponential nonlinearity, Richardson extrapolationAbstract
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 sucient 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 specied by Richardson's mesh renement method. The time interval for numerical modelling is chosen according to the decay moment's analytical upper bound. In return, numerical calculations rene the moment and the spacetime pattern of the decay. Thus, analytical and numerical parts of the investigation amplify each other.References
Korpusov M.O. Critical Exponents of Instantaneous Blow-up or Local Solvability of NonLinear Equations of Sobolev Type. Izvestiya: Mathematics, 2015, vol. 79, no. 5, pp. 955-1012. DOI: 10.1070/IM2015v079n05ABEH002768
Korpusov, M.O. Blow-up for One Sobolev Problem: Theoretical Approach and Numerical Analysis / M.O. Korpusov, D.V. Lukyanenko, A.A. Panin, E.V. Yushkov // Journal of Mathematical Analysis and Applications. - 2016. - V. 442, № 2. - P. 451-468.
Korpusov, M.O. Blow-up Phenomena in the Model of a Space Charge Stratification in Semiconductors: Analytical and Numerical Analysis / M.O. Korpusov, D.V. Lukyanenko, A.A. Panin, E.V. Yushkov // Mathematical Methods in the Applied Sciences. - 2017. - V. 40, № 7. - P. 2336-2346.
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
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
Hairer, E. Solving of Ordinary Differential Equations / E. Hairer, G. Wanner. - Berlin; Heidelberg: Spinger-Verlag, 2002.
Kalitkin N.N. [Numerical Methods for Solving Sti Systems]. Matematicheskoe modelirovanie, 1995, vol. 7, no. 5, pp. 8-11. (in Russian)
Rosenbrock, H.H. Some General Implicit Processes for the Numerical Solution of Differential Equations / H.H. Rosenbrock // The Computer Journal. - 1963. -V. 5, № 4. - P. 329-330.
Al'shin A.B., Al'shina E.A., Kalitkin N.N., Koryagina A.B. Rosenbrock Schemes with Complex Coeffcients for Stiff and Differential-Algebraic Systems. Computational Mathematics and Mathematical Physics, 2006, vol. 46, no. 8, pp. 1320-1340.
DOI: 10.1134/S0965542506080057
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.
Al'shin A.B. Numerical Diagnosis of Blow-up of Solutions of Pseudoparabolic Equations / A.B. Al'shin, E.A. Al'shina // Journal of Mathematical Sciences. - 2008. - V. 148, № 1. - C. 143-162.
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.
