Abstract
The authors study the problem of existence, uniqueness and asymptotic behavior for t → ∞ of (weak or strong) solutions of equations in the form
with various boundary and initial conditions on u(x, t). The case λ > 0 corresponds to a nonlinear Voigt model (for σ i nonlinear). The case λ = 0, N = 1 and f(u, u t ) = |ut|∝ sgn u t 0 < ∝ < 1 with nonhomogeneous boundary conditions corresponds to the motion of a linearly elastic rod in a nonlinearly viscous medium. The method followed is the Galerkin method. On leave from Hochiminh City University, Hochiminh City, Viet Nam.
Résumé
En suivant la méthode de Gaberkin, les auteurs ont étudié le probléme de l’existence, de l’unicité et du comportement asymptotique lorsque t → ∞ des solutions des équations d’ état des ondes visco-élastiques, pour diverses conditions initiales et aux limites de U (x, t). On analyse les cas auxquels correspondent des valeurs positives (modèle non linéaire de Voigt) ou nulle du paramétre, ce dernier cas étant représentatif du mouvement d’un barreau élastique linéaire dans un milieu visqueux non linéaire, monyennant l’adoption de diverses conditions aux limites.
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References
Dang Dinh Ang and A. Pham Ngoc Dinh, SIAM Journal on Mathematical Analysis 19 (1988) 337–347.
Dang Dinh Ang and A. Pham Ngoc Dinh, On the strongly damped wave equation: u u − Δu − Δu t + f(u) = 0, 19 (1988).
Dang Ding Anh and A. Pham Ngoc Dinh, Nonlinear Analysis T.M.A. 12 (1988) 581–592.
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A. Pham Ngoc Dinh and Nguyen Thanh Long, Demonstratio Mathematica 19 (1986) 45–63.
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© 1989 Kluwer Academic Publishers, Dordrecht
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Ang, D.D., Pham Ngoc Dinh, A. (1989). Some viscoelastic wave equations. In: Folias, E.S. (eds) Structural Integrity. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0927-4_4
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DOI: https://doi.org/10.1007/978-94-009-0927-4_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6906-9
Online ISBN: 978-94-009-0927-4
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