GENERAL ENERGY DECAY FOR A DEGENERATE VISCOELASTIC PETROVSKY-TYPE PLATE EQUATION WITH BOUNDARY FEEDBACK

Fushan Li; Guangwei Du

doi:10.11948/2018.390

2018 Volume 8 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(1): 390-401

Next Article Previous Article

Fushan Li, Guangwei Du. GENERAL ENERGY DECAY FOR A DEGENERATE VISCOELASTIC PETROVSKY-TYPE PLATE EQUATION WITH BOUNDARY FEEDBACK[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 390-401. doi: 10.11948/2018.390

Citation:

Fushan Li, Guangwei Du. GENERAL ENERGY DECAY FOR A DEGENERATE VISCOELASTIC PETROVSKY-TYPE PLATE EQUATION WITH BOUNDARY FEEDBACK[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 390-401. doi: 10.11948/2018.390

GENERAL ENERGY DECAY FOR A DEGENERATE VISCOELASTIC PETROVSKY-TYPE PLATE EQUATION WITH BOUNDARY FEEDBACK

Fushan Li¹,
Guangwei Du²

1 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China;
2 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129, China

Fund Project:

Abstract

Abstract

In this paper, we consider a degenerate viscoelastic Petrovsky-type plate equation K(x)u_tt + ∆²u -∫₀^t g(t -s)∆²u(s)ds + f(u)=0 with boundary feedback. Under the weaker assumption on the relaxation function, the general energy decay is proved by priori estimates and analysis of Lyapunov-like functional. The exponential decay result and polynomial decay result in some literature are special cases of this paper.
- General energy decay /
- degenerate Petrovsky plate equation /
- boundary feedback /
- function approximation
MSC: 35L05;35L15;35L70

FullText(HTML)

References

[1]	L. C. Evans, Partial Differential Equations, in:Grad. Stud. Math., vol.19, Amer. Math. Soc., Providence, RI, 1998. Google Scholar
[2]	Q. Gao, F. Li and Y. Wang, Blow-up of the solution for higher-order Kirchhofftype equations with nonlinear dissipation, Cent. Eur. J. Math., 2011, 9(3), 686-698. Google Scholar
[3]	M. A. Horn and I. Lasiecka, Uniform decay weak solutions to a von Kármán plate with nonlinear boundary dissipation, Differential and Integral Equations, 1994, 7, 885-908. Google Scholar
[4]	M. A. Horn and I. Lasiecka, Global stabilization of a dynamic von Kármán plate with nonlinear boundary feedback, Appl. Math. Optim., 1995, 31, 57-84. Google Scholar
[5]	J. R. Kang, Energy decay rates for von Kármán system with memory and boundary feedback, Appl. Math. Comput., 2012, 218, 9085-9094. Google Scholar
[6]	J. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, 10. SIAM, Philadelphia, PA, 1989. Google Scholar
[7]	F. Li and Q. Gao,Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Comput., 2016, 274, 383-392. Google Scholar
[8]	X. Lin and F. Li, Asymptotic energy estimates for nonlinear Petrovsky plate model subject to viscoelastic damping, Abstr. Appl. Anal., Volume 2012, Article ID 419717, 25 pages. Google Scholar
[9]	F. Li, Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerre-von Kármán shallow shell equations, Acta. Math. Sini., 2009, 25, 2133-2156. Google Scholar
[10]	F. Li and Y. Bai, Uniform decay rates for nonlinear viscoelastic Marguerre-von Kármán equations, J. Math. Anal. Appl., 2009, 351, 522-535. Google Scholar
[11]	F. Li, Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Kármán shallow shell system, J. Diff. Equa., 2010, 249, 1241-1257. Google Scholar
[12]	F. Li, Z. Zhao and Y. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal.:Real World Applications, 2011, 12, 1770-1784. Google Scholar
[13]	F. Li and C. Zhao, Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping, Nonlinear Anal., 2011, 74, 3468-3477. Google Scholar
[14]	F. Li and Y. Bao, Uniform Stability of the Solution for a Memory-Type Elasticity System with Nonhomogeneous Boundary Control Condition, J. Dyn. Control. Syst., 2017, 23, 301-315. Google Scholar
[15]	F. Li and J. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions, J. Math. Anal. Appl., 2012, 385, 1005-1014. Google Scholar
[16]	F. Li and J Li, Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions, Boundary Value Problems 2014, 2014:219. Google Scholar
[17]	C. A. Raposo and M. L. Santos,General decay to a von Kármán system with memory, Nonlinear Anal., 2011, 74, 937-945. Google Scholar
[18]	M. L. Santos and F. Junior, A boundary condition with memory for Kirchhoff plates equations, Appl. Math. Comput., 2004, 148, 475-496. Google Scholar
[19]	S. Y. Shin and J. R. Kang, General decay for the degenerate equation with a memory condition at the boundary, Abstr. Appl. Anal., Volume 2013, Article ID 682061, 8 pages. Google Scholar
[20]	Y. You, Energy decay and exact controllability for the Petrovsky equation in a bounded domain, Adv. Appl. Math., 1990, 11, 372-388. Google Scholar