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On a shock problem involving a nonlinear viscoelastic bar

Boundary Value Problems volume 2005, Article number: 718156 (2005) Cite this article

Abstract

We treat an initial boundary value problem for a nonlinear wave equation in the domain,. The boundary condition at the boundary point of the domain for a solution involves a time convolution term of the boundary value of at, whereas the boundary condition at the other boundary point is of the form with and given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution of this problem up to order in two small parameters,.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University HoChiMinh City, 227 Nguyen Van Cu Street, Dist.5, HoChiMinh City, Vietnam

    Nguyen Thanh Long & Tran Ngoc Diem

  2. Laboratoire de Mathématiques et Applications, physique Mathématique d'Orléans (MAPMO), UMR 6628, Bâtiment de Mathématiques, Université d'Orléans, Orléans Cedex 2, BP 6759, France

    Alain Pham Ngoc Dinh

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  1. Nguyen Thanh Long
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  2. Alain Pham Ngoc Dinh
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  3. Tran Ngoc Diem
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Correspondence to Nguyen Thanh Long.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Long, N.T., Dinh, A.P.N. & Diem, T.N. On a shock problem involving a nonlinear viscoelastic bar. Bound Value Probl 2005, 718156 (2005). https://doi.org/10.1155/BVP.2005.337

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  • DOI: https://doi.org/10.1155/BVP.2005.337