Abstract
We propose a method for analyzing the Cauchy problem for a wide class of equations with power-like nonlinearities. The method is based on the Fourier transformation, which allows reducing the original equation to an integro-differential one. We prove the existence of solutions.
Similar content being viewed by others
References
A. D. Polyanin, V. Zaitsev, and A. I. Zhurov, Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics, Fizmatlit, Moscow (2005).
V. I. Gishlarkaev, “A method for representing solutions of the Cauchy problem for linear partial differential equations,” Sb. Math., 209, 222–240 (2018).
J. D. Murray, Lectures on Nonlinear Differential Equation Models in Biology, Oxford Univ. Press, Oxford (1977).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self-Adjointness, Academic Press, New York–London (1975).
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I, Distribution Theory and Fourier Analysis (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 256), Springer, Berlin (1983).
E. F. Beckenbach and R. Bellman, An Introduction to Inequalities (New Mathematical Library, Vol. 3), Random House, New York (1961).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 207, pp. 361-375 https://doi.org/10.4213/tmf10036.
Rights and permissions
About this article
Cite this article
Gishlarkaev, V.I. Fourier transformation method for some types of nonlinear partial differential equations. Theor Math Phys 207, 713–726 (2021). https://doi.org/10.1134/S0040577921060039
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577921060039