Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation

Yongqin Liu, Shuichi Kawashima

Research output: Contribution to journalArticle

27 Citations (Scopus)

Abstract

In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space (n ≥ 2). This equation verifies the decay property of the regularity-loss type, which causes the difficulty in deriving the global a priori estimates of solutions. We overcome this difficulty by employing a time-weighted L2 energy method which makes use of the integrability of ||(δ2xu t3xu)(t)||L∞. This L∞ norm can be controlled by showing the optimal L2 decay estimates for lower-order derivatives of solutions. Thus we obtain the desired a priori estimate which enables us to prove the global existence and asymptotic decay of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is given explicitly in terms of the fundamental solution of a fourth-order linear parabolic equation.

Original languageEnglish
Pages (from-to)1113-1139
Number of pages27
JournalDiscrete and Continuous Dynamical Systems
Volume29
Issue number3
DOIs
Publication statusPublished - Mar 1 2011

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Plate Equation
Dissipative Equations
Asymptotic Behavior of Solutions
A Priori Estimates
Global Existence
Regularity
Decay of Solutions
Decay Estimates
Initial value problems
Energy Method
Fundamental Solution
Integrability
Parabolic Equation
Initial Value Problem
Fourth Order
Linear equation
Decay
Verify
Derivatives
Norm

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. / Liu, Yongqin; Kawashima, Shuichi.

In: Discrete and Continuous Dynamical Systems, Vol. 29, No. 3, 01.03.2011, p. 1113-1139.

Research output: Contribution to journalArticle

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