### Abstract

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

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

Pages (from-to) | 471-501 |

Number of pages | 31 |

Journal | Journal of Hyperbolic Differential Equations |

Volume | 7 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Sep 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- decay estimates
- Dissipative plate equation
- global existence
- regularity-loss property

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

### Cite this

**Decay estimates of solutions to a semi-linear dissipative plate equation.** / Sugitani, Yousuke; Kawashima, Shuichi.

Research output: Contribution to journal › Article

*Journal of Hyperbolic Differential Equations*, vol. 7, no. 3, pp. 471-501. https://doi.org/10.1142/S0219891610002207

}

TY - JOUR

T1 - Decay estimates of solutions to a semi-linear dissipative plate equation

AU - Sugitani, Yousuke

AU - Kawashima, Shuichi

PY - 2010/9/1

Y1 - 2010/9/1

N2 - We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

AB - We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

KW - decay estimates

KW - Dissipative plate equation

KW - global existence

KW - regularity-loss property

UR - http://www.scopus.com/inward/record.url?scp=78049372466&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78049372466&partnerID=8YFLogxK

U2 - 10.1142/S0219891610002207

DO - 10.1142/S0219891610002207

M3 - Article

AN - SCOPUS:78049372466

VL - 7

SP - 471

EP - 501

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

SN - 0219-8916

IS - 3

ER -