Global existence and decay of solutions for a quasi-linear dissipative plate equation

Yongqin Liu, Shuichi Kawashima

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2 energy method combined with the optimal L2 decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates 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 the fundamental solution of the corresponding fourth-order linear parabolic equation.

Original languageEnglish
Pages (from-to)591-614
Number of pages24
JournalJournal of Hyperbolic Differential Equations
Volume8
Issue number3
DOIs
Publication statusPublished - 2011 Sep 1
Externally publishedYes

Fingerprint

Plate Equation
Decay of Solutions
Dissipative Equations
Global Existence
Existence of Solutions
Decay Estimates
Regularity
Energy Method
Fundamental Solution
Parabolic Equation
Initial Value Problem
Fourth Order
Linear equation
Exponent
Decay
Verify
Derivative
Arbitrary

Keywords

  • asymptotic behavior
  • decay estimates
  • global existence
  • Quasi-linear dissipative plate equation
  • time-weighted energy method

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis

Cite this

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

In: Journal of Hyperbolic Differential Equations, Vol. 8, No. 3, 01.09.2011, p. 591-614.

Research output: Contribution to journalArticle

@article{6537f4ae5fb946b19e2b478b077eb5ce,
title = "Global existence and decay of solutions for a quasi-linear dissipative plate equation",
abstract = "In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2 energy method combined with the optimal L2 decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates 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 the fundamental solution of the corresponding fourth-order linear parabolic equation.",
keywords = "asymptotic behavior, decay estimates, global existence, Quasi-linear dissipative plate equation, time-weighted energy method",
author = "Yongqin Liu and Shuichi Kawashima",
year = "2011",
month = "9",
day = "1",
doi = "10.1142/S0219891611002500",
language = "English",
volume = "8",
pages = "591--614",
journal = "Journal of Hyperbolic Differential Equations",
issn = "0219-8916",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "3",

}

TY - JOUR

T1 - Global existence and decay of solutions for a quasi-linear dissipative plate equation

AU - Liu, Yongqin

AU - Kawashima, Shuichi

PY - 2011/9/1

Y1 - 2011/9/1

N2 - In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2 energy method combined with the optimal L2 decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates 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 the fundamental solution of the corresponding fourth-order linear parabolic equation.

AB - In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2 energy method combined with the optimal L2 decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates 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 the fundamental solution of the corresponding fourth-order linear parabolic equation.

KW - asymptotic behavior

KW - decay estimates

KW - global existence

KW - Quasi-linear dissipative plate equation

KW - time-weighted energy method

UR - http://www.scopus.com/inward/record.url?scp=80052587628&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052587628&partnerID=8YFLogxK

U2 - 10.1142/S0219891611002500

DO - 10.1142/S0219891611002500

M3 - Article

VL - 8

SP - 591

EP - 614

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

SN - 0219-8916

IS - 3

ER -