Abstract
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).
| 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.
In: Evolution Equations and Control Theory, Vol. 2, No. 1, 2013, p. 101-117.Research output: Contribution to journal › Article
}
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 -