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A shock problem involving a nonlinear viscoelastic bar associated with a nonlinear boundary condition

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We study the initial-boundary value problem for a nonlinear wave equation given by [...] where p > 2, q > l, K, lambda are given constants and uo, u1, F are given functions, the unknown function u(x,t) and the unknown boundary value P (t) satisfy the following nonlinear integral equation [...] where K1, alpha, beta are given constants and g, k arę given functions. In Part 1 we prove a theorem of existence and uniqueness of a weak solution (u, P) of problem (1), (2). The proof is based on the Faedo-Galerkin method associated with a priori estimates, weak convergence and compactness techniques. In Part 3 we obtain an asymptotic expansion of the solution (u, P) of the problem (1), (2) up to order N+1 in three small parameters K, lambda, K1.
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Bibliogr. 12 poz.
  • Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Hochiminh City, 227 Nguyen Van Cu Str., Dist. 5, Hochiminh City, Vietnam,
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  • [9] N. T. Long, A. P. N. Dinh, T. N. Diem, On a shock problem involving a nonlinear viscoelastic bar, J. Boundary Value Problems, Hindawi Publishing Corporation, 2005 (3) (2005), 337-358.
  • [10] N. T. Long, L. V. Ut, N. T. T. Truc, On a shock problem involving a linear viscoelastic bar, Nonlinear Anal. 63 (2) (2005), 198-224.
  • [11] N. T. Long, V. G. Giai, A wave equation associated with mixed nonhomogeneous conditions: Global existence and asymptotic expansion of solutions, Nonlinear Anal. 66 (7) (2007), 1526-1546.
  • [12] N. T. Long, L. X. Truong, Existence and asymptotic espansion for a mscoelastic problem with a mixed nonhomogeneous condition, Nonlinear Anal. 67 (3) (2007), 842-864.
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