Abstract
The authors study the problem of existence, uniqueness and asymptotic behavior for t→∞ of (weak or strong) solutions of equations in the form
% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% abaeqabaGaamyDamaaBaaaleaacaWG0bGaamiDaaqabaGccqGHsisl% cqaH7oaBcqqHuoarcaWG1bWaaSbaaSqaaiaadshaaeqaaOGaeyOeI0% YaaabCaeaacqGHciITcaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWG% Pbaabeaakiabeo8aZnaaBaaaleaacaWGPbaabeaakiaacIcacaWG1b% WaaSbaaSqaaiaadIhadaWgaaadbaGaamyAaaqabaaaleqaaOGaaiyk% aiabgUcaRaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGobaaniabgg% HiLdGccaWGMbGaaiikaiaadwhacaGGSaGaamyDamaaBaaaleaacaWG% 0baabeaakiaacMcacaqGGaGaeyypa0Jaaeiiaiaaicdaaeaaaeaaae% aaaeaacqaH7oaBcaqGGaGaeyyzImRaaeiiaiaaicdacaqGGaGaaiik% aiaadIhacaGGSaGaamiDaiaacMcacqGHiiIZcqGHPoWvcaqGGaGaae% iEaiaabccacaGGOaGaaGimaiaacYcacaWGubGaaiykaaqaaaqaaaqa% aaqaaiabgM6axjaabccacaqG9aGaaeiiaiaabggacaqGGaGaaeizai% aab+gacaqGTbGaaeyyaiaabMgacaqGUbGaaeiiaiaabMgacaqGUbGa% aeiiaiabl2riHoaaCaaaleqabaGaamOBaaaakiaacYcaaaaa!879A!\[\begin{array}{l}u_{tt} - \lambda \Delta u_t - \sum\limits_{i = 1}^N {\partial /\partial x_i \sigma _i (u_{x_i } ) + } f(u,u_t ){\rm{ }} = {\rm{ }}0 \\\\\\\\\lambda {\rm{ }} \ge {\rm{ }}0{\rm{ }}(x,t) \in \Omega {\rm{ x }}(0,T) \\\\\\\\\Omega {\rm{ = a domain in }}^n , \\\end{array}\]with various boundary and initial conditions on u(x, t). The case λ>0 corresponds to a nonlinear Voigt model (for σ nonlinear). The case λ=0, N=1 and f(u, u 1 )=|u 1 |α sgn u 1 , 0<α<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.
Résumé
En suivant la méthode de Gaberkin, les auteurs ont étudié le problème 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 paramètre, 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
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Ang, D.D., Pham Ngoc Dinh, A. Some viscoelastic wave equations. Int J Fract 39, 35–43 (1989). https://doi.org/10.1007/BF00047438
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DOI: https://doi.org/10.1007/BF00047438