### Abstract

It has been asserted that the damped wave equation has the diffusive structure as t → ∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the L^{p} - L^{q} estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the L^{p} - L^{q} estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained.

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

Pages (from-to) | 631-649 |

Number of pages | 19 |

Journal | Mathematische Zeitschrift |

Volume | 244 |

Issue number | 3 |

Publication status | Published - 2003 Jul |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{p}-L

^{q}estimates of solutions to the damped wave equation in 3-dimensional space and their application.

*Mathematische Zeitschrift*,

*244*(3), 631-649.

**L ^{p}-L^{q} estimates of solutions to the damped wave equation in 3-dimensional space and their application.** / Nishihara, Kenji.

Research output: Contribution to journal › Article

^{p}-L

^{q}estimates of solutions to the damped wave equation in 3-dimensional space and their application',

*Mathematische Zeitschrift*, vol. 244, no. 3, pp. 631-649.

^{p}-L

^{q}estimates of solutions to the damped wave equation in 3-dimensional space and their application. Mathematische Zeitschrift. 2003 Jul;244(3):631-649.

}

TY - JOUR

T1 - Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application

AU - Nishihara, Kenji

PY - 2003/7

Y1 - 2003/7

N2 - It has been asserted that the damped wave equation has the diffusive structure as t → ∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the Lp - Lq estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the Lp - Lq estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained.

AB - It has been asserted that the damped wave equation has the diffusive structure as t → ∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the Lp - Lq estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the Lp - Lq estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained.

UR - http://www.scopus.com/inward/record.url?scp=0043245941&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0043245941&partnerID=8YFLogxK

M3 - Article

VL - 244

SP - 631

EP - 649

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3

ER -