| Title:
|
Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation (English) |
| Author:
|
Park, Jong Yeoul |
| Author:
|
Park, Sun Hye |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
273-286 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
asymptotic stability |
| Keyword:
|
viscoelastic problems |
| Keyword:
|
boundary dissipation |
| Keyword:
|
wave equation |
| MSC:
|
35B35 |
| MSC:
|
35B40 |
| MSC:
|
35L15 |
| MSC:
|
35L70 |
| MSC:
|
35Q72 |
| MSC:
|
65M60 |
| MSC:
|
74D10 |
| MSC:
|
74H20 |
| idZBL:
|
Zbl 1164.35445 |
| idMR:
|
MR2291736 |
| . |
| Date available:
|
2009-09-24T11:33:00Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128066 |
| . |
| Reference:
|
[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. MR 1950077, 10.1016/S0096-3003(02)00163-7 |
| Reference:
|
[2] E. E. Beckenbach and R. Bellman: Inequalities.Springer-Verlag, Berlin, 1971. MR 0192009 |
| Reference:
|
[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. Zbl 1009.74034, MR 1897875 |
| Reference:
|
[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 1893844 |
| Reference:
|
[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 1797934 |
| Reference:
|
[6] I. Lasiecka, D. Tataru: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping.Diff. Int. Eqs. 6 (1993), 507–533. MR 1202555 |
| Reference:
|
[7] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod-Gauthier Villars, Paris, 1969. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[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. MR 1897321, 10.1016/S0096-3003(01)00031-5 |
| Reference:
|
[9] B. Rao: Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear feedback.Nonlinear Anal., T.M.A. 20 (1993), 605–626. MR 1214731, 10.1016/0362-546X(93)90023-L |
| Reference:
|
[10] M. L. Santos: Decay rates for solutions of a system of wave equations with memory.Elect. J. Diff. Eqs. 38 (2002), 1–17. Zbl 1010.35012, MR 1907714 |
| Reference:
|
[11] E. Zuazua: Uniform stabilization of the wave equation by nonlinear boundary feedback.SIAM J. Control Optim. 28 (1990), 466–477. Zbl 0695.93090, MR 1040470, 10.1137/0328025 |
| . |
