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Study of the classical solution to the one-dimensional mixed problem for a class of semilinear long-wave equations

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Abstract

Many problems in mathematical physics are reduced to one- or multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear Sobolev-type equations. In this work, local and global classical solvability is studied for the one-dimensional mixed problem with homogeneous Riquier-type boundary conditions for a class of semilinear long-wave equations

$$ U_{tt} \left( {t,x} \right) - U_{xx} \left( {t,x} \right) - \alpha U_{ttxx} \left( {t,x} \right) = F\left( {t,x,U\left( {t,x} \right),U_x \left( {t,x} \right),U_{xx} \left( {t,x} \right),U_t \left( {t,x} \right),U_{tx} \left( {t,x} \right),U_{txx} \left( {t,x} \right)} \right) $$

, where α > 0 is a fixed number, 0 ≤ t ≤ T, 0 ≤ x ≤ π, 0 < T < +∞, F is a given function, and U(t, x) is the sought function. A uniqueness theorem for the mixed problem is proved using the Gronwall-Bellman inequality. A local existence result is proved by applying the generalized contraction mapping principle combined with the Schauder fixed point theorem. The method of a priori estimates is used to prove the global existence of a classical solution to the mixed problem.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Baku State University, ul. Z. Khalilova 23, Baku, AZ1148, Azerbaijan

    F. M. Namazov & K. I. Khudaverdiyev

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  1. F. M. Namazov
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  2. K. I. Khudaverdiyev
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Correspondence to K. I. Khudaverdiyev.

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Original Russian Text © F.M. Namazov, K.I. Khudaverdiyev, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 9, pp. 1569–1586.

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Namazov, F.M., Khudaverdiyev, K.I. Study of the classical solution to the one-dimensional mixed problem for a class of semilinear long-wave equations. Comput. Math. and Math. Phys. 50, 1494–1510 (2010). https://doi.org/10.1134/S0965542510090034

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  • DOI: https://doi.org/10.1134/S0965542510090034

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