### Abstract

We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)u_{t} and absorbed semilinear term {pipe}u{pipe}^{ρ-1}u in R^{N}. Our assumption on V(x) ~ (1 + {pipe}x{pipr}^{2})^{-α/2} (0 ≤ α < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρ_{c}(N, α):=1+2/(N - α) and for ρ < ρ_{c}(N, α). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρ_{c}(N, α) is a critical exponent. Note that ρ_{c}(N, α) with α = 0 coincides to the Fujita exponent ρ_{F}(N):=1+2/N.

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

Pages (from-to) | 1402-1418 |

Number of pages | 17 |

Journal | Communications in Partial Differential Equations |

Volume | 35 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2010 Aug |

### Fingerprint

### Keywords

- Absorbed semilinear term
- Damped wave equation
- Space dependent potential

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term.** / Nishihara, Kenji.

Research output: Contribution to journal › Article

*Communications in Partial Differential Equations*, vol. 35, no. 8, pp. 1402-1418. https://doi.org/10.1080/03605302.2010.490285

}

TY - JOUR

T1 - Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term

AU - Nishihara, Kenji

PY - 2010/8

Y1 - 2010/8

N2 - We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)ut and absorbed semilinear term {pipe}u{pipe}ρ-1u in RN. Our assumption on V(x) ~ (1 + {pipe}x{pipr}2)-α/2 (0 ≤ α < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρc(N, α):=1+2/(N - α) and for ρ < ρc(N, α). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρc(N, α) is a critical exponent. Note that ρc(N, α) with α = 0 coincides to the Fujita exponent ρF(N):=1+2/N.

AB - We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)ut and absorbed semilinear term {pipe}u{pipe}ρ-1u in RN. Our assumption on V(x) ~ (1 + {pipe}x{pipr}2)-α/2 (0 ≤ α < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρc(N, α):=1+2/(N - α) and for ρ < ρc(N, α). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρc(N, α) is a critical exponent. Note that ρc(N, α) with α = 0 coincides to the Fujita exponent ρF(N):=1+2/N.

KW - Absorbed semilinear term

KW - Damped wave equation

KW - Space dependent potential

UR - http://www.scopus.com/inward/record.url?scp=77954296885&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954296885&partnerID=8YFLogxK

U2 - 10.1080/03605302.2010.490285

DO - 10.1080/03605302.2010.490285

M3 - Article

VL - 35

SP - 1402

EP - 1418

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 8

ER -