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) L 2 energy method combined with the optimal L 2 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.

Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation

“…For more higher-order wave equations, we refer to [23][24][25][26]21] and the references therein. Also we refer to [7][8][9]17] for various aspects of dissipation of the plate equation.…”

New Stability Criterion for the Dissipative Linear System and Analysis of Bresse System

“…In recent 10 years, some complicated physical models which possess the weak dissipative structure called the regularity-loss structure was studied. For example, the dissipative Timoshenko system was discussed in [8][9][10], the Euler-Maxwell system was studied in [11,12], and the hybrid problem of plate equations is in [13][14][15][16]. We would like to emphasize that these physical models do not satisfy (4) but Condition (A).…”

“…where P jk , Q j and R arem × m real constant matrices. In fact, a lot of physical models are described as (1) under (14). For example, the linearized system of the electro-magneto-fluid dynamics and Euler-Maxwell system are described as (1) under (14).…”

“…In fact, a lot of physical models are described as (1) under (14). For example, the linearized system of the electro-magneto-fluid dynamics and Euler-Maxwell system are described as (1) under (14). For the detail, we refer [2,12] to the reader.…”

Decay property of regularity-loss type for the Euler-Maxwell system

“…A similar dissipative structure of the regularity-loss type was also found for various systems of partial differential equations. See [16,8,9] for the dissipative Timoshenko system, [7,13] for hyperbolic-elliptic systems related to the radiation hydrodynamics, and [4,21,14,15] for a dissipative plate equation.…”