TY - JOUR

T1 - Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation

AU - Liu, Yongqin

AU - Kawashima, Shuichi

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/3

Y1 - 2011/3

N2 - 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 t,δ3xu)(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.

AB - 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 t,δ3xu)(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.

KW - Asymptotic behavior

KW - Decay estimates

KW - Global existence

KW - Quasi-linear dissipative plate equation

KW - Timeweighted energy method

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U2 - 10.3934/dcds.2011.29.1113

DO - 10.3934/dcds.2011.29.1113

M3 - Article

AN - SCOPUS:79952248749

VL - 29

SP - 1113

EP - 1139

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 3

ER -