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

• Autorzy: Long N. , Giai V. , Truong L.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Faedo-Galerkin method; nonlinear wave equation; asymptotic expansion; existence and uniqueness

PL

metoda Faedo-Galerkina; rozkład asymptotyczny; równania nieliniowe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2008
• Tom: Vol. 41, nr 1
• Strony: 85--108
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Long N.

autor

Giai V.

autor

Truong L.

• 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,

Bibliografia

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• [3] M. Bergounioux, N. T. Long, A. P. N. Dinh, Mathematical model for a shock problem involving a linear viscoelastic bar, Nonlinear Anal. 43 (5) (2001), 547-561.
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• [5] E. L. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955, p. 43.
• [6] J. L. Lions, W. A. Strauss, Some nonlinear evolution equationtions, Bull. Soc. Math., Fiance, 93 (1965), 43-96.
• [7] J. L. Lions, Equations differentielles operationelles et problemes aux limites, Springer-Verlag, Berlin, 1961, p. 96, Section 7.
• [8] J. L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod; Gauthier-Villars, Paris, 1969.
• [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.