Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function

In the strip Π = (-1, 0) × ℝ, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u t + u xxx + uu x = 0 with initial condition either 1) u(-1, x) = -xθ(x), or 2) u(-1, x) = -xθ(-x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (-1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity. © 2008 Pleiades Publishing, Ltd.

Authors
Editors
-
Publisher
-
Number of issue
1-2
Language
English
Pages
107-115
Status
Published
Department
-
Number
-
Volume
83
Year
2008
Organizations
  • 1 Russian Peoples' Friendship University, Moscow, Russian Federation
Keywords
Banach space; Bochner measurable mapping; Burgers equation; Cauchy problem; Gas-dynamic problem; Korteweg-de Vries equation; Line of weak discontinuity
Date of creation
19.10.2018
Date of change
19.10.2018
Short link
https://repository.rudn.ru/en/records/article/record/3136/