danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2023-67-5-366-372dan-1149Research ArticleМАТЕМАТИКАMATHEMATICSТеоремы сравнения в задачах разрушения решения для уравнения реакции диффузии и в их аппроксимацияхComparison theorem in blow-up problems for reaction diffusion equations and for their approximationsЩадинскийД. А.SchadinskiiD. A.

Денис Александрович Щадинский, мл. науч. сотрудник, магистр физ.-мат. наук

220072

ул. Сурганова, 11

Минск

Denis A. Schadinskii, Junior Researcher, Master (Physics and Mathematics)

220072

11, Surganov Str.

Minsk

schadinskii@gmail.comИнститут математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus202330102023675366372Copyright © Щадинский Д.А., 20232023Щадинский Д.А.Schadinskii D.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1149

   Найдены достаточные условия разрушения решения и верхняя оценка времени разрушения решения для задач Неймана и Дирихле для уравнения реакции диффузии с градиентной нелинейностью, которые получены на основе теорем сравнения, неравенства Йенсена и законов сохранения. Используя аналогичную технику доказательства для разностного случая, была построена разностная схема, аппроксимирующая ранее упомянутую задачу Неймана, для которой получены достаточные условия разрушения решения и оценка времени разрушения решения, согласованные с соответствующими условиями и оценками для дифференциального случая.

   In this paper, blow-up sufficient conditions and upper bound of blow-up time for solution of Neumann and Dirichlet problems for reaction diffusion equations with non-linear gradient have been obtained. These equations have been found from the comparison of theorems, Jensen’s inequality and conservations laws. By using a similar proof approach for the finite-difference case, the finite-difference scheme was constructed, approximating the above-mentioned Neumann problem, for which sufficient conditions and upper bound of blow-up time, consistent with appropriate conditions and bound for the appropriate differential problem, have been obtained.

теоремы сравненияразрушение решениядискретные аналоги теорем сравненияуравнения реакции диффузииcomparison theoremsblow-updiscrete analogue of comparison theoremsreaction diffusion equationsReferencesBlow-up in quasilinear parabolic equations / A. A. Samarskii [et al.]. – Berlin, 1995. – 560 p. doi: 10.1515/9783110889864Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P. Blow-up in quasilinear parabolic equations. Berlin, 1995. 560 p. doi: 10.1515/9783110889864Matus, P. P. On the Role of Conservation Laws and Input Data in the Generation of Peaking Modes in Quasilinear Multidimensional Parabolic Equations with Nonlinear Source and in their Approximations / P. P. Matus, N. G. Churbanova, D. A. Shchadinskii // Differ. Equ. – 2016. – Vol. 52, N 7. – P. 942–950. doi: 10.1134/s0012266116070120Matus P. P., Churbanova N. G., Shchadinskii D. A. On the Role of Conservation Laws and Input Data in the Generation of Peaking Modes in Quasilinear Multidimensional Parabolic Equations with Nonlinear Source and in their Approximations. Differential Equations, 2016, vol. 52, no. 7, pp. 942–950. doi: 10.1134/s0012266116070120Matus, P. P. On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations / P. P. Matus // Differ. Equ. – 2013. – Vol. 49, N 7. – P. 883–894. doi: 10.1134/S0012266113070100Matus P. P. On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations. Differential Equations, 2013, vol. 49, no. 7, pp. 883–894. doi: 10.1134/s0012266113070100Discrete Analogs of the Comparison Theorem and Two-Sided Estimates of Solution of Parabolic Equations / P. Matus [et al.] // Appl. Math. Inf. Sci. – 2016. – Vol. 10, N 1. – P. 83–92. doi: 10.18576/amis/100108Matus P., Kozera R., Paradzinska A., Schadinskii D. Discrete Analogs of the Comparison Theorem and Two-Sided Estimates of Solution of Parabolic Equations. Applied Mathematics & Information Sciences, 2016, vol. 10, no. 1, pp. 83–92. doi: 10.18576/amis/100108Bandle, C. Blow-up in diffusion equations: A survey / C. Bandle, H. Brunner // J. Comput. Appl. Math. – 1998. – Vol. 97, N 1–2. – P. 3–22. doi: 10.1016/s0377-0427(98)00100-9Bandle C., Brunner H. Blow-up in diffusion equations: A survey. Journal of Computational and Applied Mathematics, 1998, vol. 97, no. 1–2, pp. 3–22. doi: 10.1016/s0377-0427(98)00100-9Straughan, B. Explosive Instabilities in Mechanics / B. Straughan. – Berlin, 1998. – 197 p. doi: 10.1007/978-3-642-58807-5Straughan B. Explosive Instabilities in Mechanics. Berlin, 1998. 197 p. doi: 10.1007/978-3-642-58807-5Ding, J. Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions / J. Ding, H. Hu // J. Math. Anal. Appl. – 2016. – Vol. 433, N 2. – P. 1718–1735. doi: 10.1016/j.jmaa.2015.08.046Ding J., Hu H. Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions. Journal of Mathematical Analysis and Applications, 2016, vol. 433, no. 2, pp. 1718–1735. doi: 10.1016/j.jmaa.2015.08.046Payne, L. E. Blow-up phenomena for some nonlinear parabolic problems / L. E. Payne, G. A. Philippin, P. W. Schaefer // Nonlinear Anal., Theory, Methods, Appl. – 2008. – Vol. 69, N 10. – P. 3495–3502. doi: 10.1016/j.na.2007.09.035Payne L. E., Philippin G. A., Schaefer P. W. Blow-up phenomena for some nonlinear parabolic problems. Nonlinear Analysis: Theory, Methods and Applications, 2008, vol. 69, no. 10, pp. 3495–3502. doi: 10.1016/j.na.2007.09.035Mu, C. Blow-up phenomena for a doubly degenerate equation with positive initial energy / C. Mu, R. Zeng, B. Chen // Nonlinear Anal., Theory, Methods, Appl. – 2010. – Vol. 72, N 2. – P. 782–793. doi: 10.1016/j.na.2009.07.020Mu C., Zeng R., Chen B. Blow-up phenomena for a doubly degenerate equation with positive initial energy. Nonlinear Analysis: Theory, Methods and Applications, 2010, vol. 72, no. 2, pp. 782–793. doi: 10.1016/j.na.2009.07.020Protter, M. H. Maximum Principles in Differential Equations / M. H. Protter, H. F. Weinberger. – Prentice Hall, 1967. – 261 p.Protter M. H., Weinberger H. F. Maximum Principles in Differential Equations. New Jersey, 1967. 261 p.

The authors declare that there are no conflicts of interest present.