Geodesic
Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 273-286
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We consider the damped semilinear viscoelastic wave equation \[ u^{\prime \prime } - \Delta u + \int ^t_0 h (t-\tau ) \div \lbrace a \nabla u(\tau ) \rbrace \mathrm{d}\tau + g(u^{\prime }) = 0 \quad \text{in}\hspace{5.0pt}\Omega \times (0,\infty ) \] with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.
We consider the damped semilinear viscoelastic wave equation \[ u^{\prime \prime } - \Delta u + \int ^t_0 h (t-\tau ) \div \lbrace a \nabla u(\tau ) \rbrace \mathrm{d}\tau + g(u^{\prime }) = 0 \quad \text{in}\hspace{5.0pt}\Omega \times (0,\infty ) \] with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.
Classification : 35B35, 35B40, 35L15, 35L70, 35Q72, 65M60, 74D10, 74H20
Keywords: asymptotic stability; viscoelastic problems; boundary dissipation; wave equation 
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Park, Jong Yeoul; Park, Sun Hye. Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 273-286. http://geodesic.mathdoc.fr/item/CMJ_2006_56_2_a0/

[1] J. J.  Bae, J. Y.  Park, and J. M.  Jeong: On uniform decay of solutions for wave equation of Kirchhoff type with nonlinear boundary damping and memory source term. Appl. Math. Comput. 138 (2003), 463–478. | DOI | MR

[2] E. E. Beckenbach and R. Bellman: Inequalities. Springer-Verlag, Berlin, 1971. | MR

[3] M. M. Cavalcanti: Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete and Continuous Dynamical Systems 8 (2002), 675–695. | MR | Zbl

[4] M. M.  Cavalcanti, V. N.  Domingos Cavalcanti, T. F.  Ma, and J. A.  Soriano: Global existence and asymptotic stability for viscoelastic problems. Differential and Integral Equations 15 (2002), 731–748. | MR

[5] M. M.  Cavalcanti, V. N.  Domingos Cavalcanti, J. S.  Prates Filho, and J. A.  Soriano: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Diff. Int. Eqs. 14 (2001), 85–116. | MR

[6] I.  Lasiecka, D.  Tataru: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Diff. Int. Eqs. 6 (1993), 507–533. | MR

[7] J.-L.  Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier Villars, Paris, 1969. | MR | Zbl

[8] J. Y.  Park, J. J.  Bae: On coupled wave equation of Kirchhoff type with nonliear boundary damping and memory term. Appl. Math. Comput. 129 (2002), 87–105. | DOI | MR

[9] B.  Rao: Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear feedback. Nonlinear Anal., T.M.A. 20 (1993), 605–626. | DOI | MR

[10] M. L.  Santos: Decay rates for solutions of a system of wave equations with memory. Elect.  J. Diff. Eqs. 38 (2002), 1–17. | MR | Zbl

[11] E.  Zuazua: Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J.  Control Optim. 28 (1990), 466–477. | DOI | MR | Zbl