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Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class L p for p ≥ 1

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Abstract

We prove that the mixed problem for the Klein–Gordon–Fock equation u tt (x, t) − u xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L p (Q T ) for p ≥ 1. We construct the solution in explicit analytic form.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    A. A. Kuleshov, I. S. Mokrousov & I. N. Smirnov

  2. Steklov Mathematical Institute, Moscow, 119991, Russia

    A. A. Kuleshov, I. S. Mokrousov & I. N. Smirnov

  3. Peoples’ Friendship University of Russia, Moscow, 117198, Russia

    A. A. Kuleshov, I. S. Mokrousov & I. N. Smirnov

Authors
  1. A. A. Kuleshov
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  2. I. S. Mokrousov
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  3. I. N. Smirnov
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Correspondence to A. A. Kuleshov.

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Original Russian Text © A.A. Kuleshov, I.S. Mokrousov, I.N. Smirnov, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 3, pp. 336–340.

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Kuleshov, A.A., Mokrousov, I.S. & Smirnov, I.N. Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class L p for p ≥ 1. Diff Equat 54, 330–334 (2018). https://doi.org/10.1134/S0012266118030059

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  • DOI: https://doi.org/10.1134/S0012266118030059

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