### Abstract

The regularity conservation as well as the smoothing effect are studied for the equation u″+Au+cA^{α}u′ = 0, where A is a positive selfadjoint operator on a real Hilbert space H and α ∈ (0, 1]; c > 0. When α ≥ 1/2 the equation generates an analytic semigroup on D(A^{1/2}) × H, and if α ∈ (0, 1/2) a weaker optimal smoothing property is established. Some conservation prop- erties in other norms are also established, as a typical example, the strongly dissipative wave equation u_{tt} – Δu – cΔu_{t} = 0 with Dirichlet boundary conditions in a bounded domain is given, for which the space C_{0}(Ω) × C_{0}(Ω) is conserved for t > 0, which presents a sharp contrast with the conservative case u_{tt} – Δu = 0 for which C0()-regularity can be lost even starting from an initial state (u_{0}, 0) with u_{0} 2 C_{0}(Ω) ⋂ C^{1}(Ω).

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

Pages (from-to) | 101-117 |

Number of pages | 17 |

Journal | Evolution Equations and Control Theory |

Volume | 2 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 |

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### Keywords

- Analytic semi-group
- Regularity
- Smoothing effect

### ASJC Scopus subject areas

- Applied Mathematics
- Control and Optimization
- Modelling and Simulation

### Cite this

**Analyticity and regularity for a class of second order evolution equations.** / Haraux, Alain; Otani, Mitsuharu.

Research output: Contribution to journal › Article

*Evolution Equations and Control Theory*, vol. 2, no. 1, pp. 101-117. https://doi.org/10.3934/eect.2013.2.101

}

TY - JOUR

T1 - Analyticity and regularity for a class of second order evolution equations

AU - Haraux, Alain

AU - Otani, Mitsuharu

PY - 2013

Y1 - 2013

N2 - The regularity conservation as well as the smoothing effect are studied for the equation u″+Au+cAαu′ = 0, where A is a positive selfadjoint operator on a real Hilbert space H and α ∈ (0, 1]; c > 0. When α ≥ 1/2 the equation generates an analytic semigroup on D(A1/2) × H, and if α ∈ (0, 1/2) a weaker optimal smoothing property is established. Some conservation prop- erties in other norms are also established, as a typical example, the strongly dissipative wave equation utt – Δu – cΔut = 0 with Dirichlet boundary conditions in a bounded domain is given, for which the space C0(Ω) × C0(Ω) is conserved for t > 0, which presents a sharp contrast with the conservative case utt – Δu = 0 for which C0()-regularity can be lost even starting from an initial state (u0, 0) with u0 2 C0(Ω) ⋂ C1(Ω).

AB - The regularity conservation as well as the smoothing effect are studied for the equation u″+Au+cAαu′ = 0, where A is a positive selfadjoint operator on a real Hilbert space H and α ∈ (0, 1]; c > 0. When α ≥ 1/2 the equation generates an analytic semigroup on D(A1/2) × H, and if α ∈ (0, 1/2) a weaker optimal smoothing property is established. Some conservation prop- erties in other norms are also established, as a typical example, the strongly dissipative wave equation utt – Δu – cΔut = 0 with Dirichlet boundary conditions in a bounded domain is given, for which the space C0(Ω) × C0(Ω) is conserved for t > 0, which presents a sharp contrast with the conservative case utt – Δu = 0 for which C0()-regularity can be lost even starting from an initial state (u0, 0) with u0 2 C0(Ω) ⋂ C1(Ω).

KW - Analytic semi-group

KW - Regularity

KW - Smoothing effect

UR - http://www.scopus.com/inward/record.url?scp=84898014112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898014112&partnerID=8YFLogxK

U2 - 10.3934/eect.2013.2.101

DO - 10.3934/eect.2013.2.101

M3 - Article

VL - 2

SP - 101

EP - 117

JO - Evolution Equations and Control Theory

JF - Evolution Equations and Control Theory

SN - 2163-2472

IS - 1

ER -