Abstract
In the strip П = (−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.
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Original Russian Text © A. V. Faminskii, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 1, pp. 119–128.
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Faminskii, A.V. Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function. Math Notes 83, 107–115 (2008). https://doi.org/10.1134/S0001434608010136
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DOI: https://doi.org/10.1134/S0001434608010136