### Abstract

In this paper we focus on the initial value problem of the semilinear plate equation with memory in multi-dimensions (n > 1), the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.

Original language | English |
---|---|

Pages (from-to) | 531-547 |

Number of pages | 17 |

Journal | Kinetic and Related Models |

Volume | 4 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modelling and Simulation

### Cite this

*Kinetic and Related Models*,

*4*(2), 531-547. https://doi.org/10.3934/krm.2011.4.531

**Decay property for a plate equation with memory-type dissipation.** / Liu, Yongqin; Kawashima, Shuichi.

Research output: Contribution to journal › Article

*Kinetic and Related Models*, vol. 4, no. 2, pp. 531-547. https://doi.org/10.3934/krm.2011.4.531

}

TY - JOUR

T1 - Decay property for a plate equation with memory-type dissipation

AU - Liu, Yongqin

AU - Kawashima, Shuichi

PY - 2011/6/1

Y1 - 2011/6/1

N2 - In this paper we focus on the initial value problem of the semilinear plate equation with memory in multi-dimensions (n > 1), the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.

AB - In this paper we focus on the initial value problem of the semilinear plate equation with memory in multi-dimensions (n > 1), the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.

UR - http://www.scopus.com/inward/record.url?scp=79957943709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957943709&partnerID=8YFLogxK

U2 - 10.3934/krm.2011.4.531

DO - 10.3934/krm.2011.4.531

M3 - Article

VL - 4

SP - 531

EP - 547

JO - Kinetic and Related Models

JF - Kinetic and Related Models

SN - 1937-5093

IS - 2

ER -