Abstract
The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space Ḣs = Ḣs(ℝn) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data (φ,ψ) ∈ Ḣ ≡ Ḣs ⊕ Ḣs-1, ∥(φ,ψ); Ḣ1/2∥ is relatively small with respect to ∥(φ,ψ); Ḣσ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u1+4/(n-1) near zero and has an arbitrary growth rate at infinity.
| Original language | English |
|---|---|
| Pages (from-to) | 613-632 |
| Number of pages | 20 |
| Journal | Houston Journal of Mathematics |
| Volume | 27 |
| Issue number | 3 |
| Publication status | Published - 2001 |
| Externally published | Yes |
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Keywords
- Besov spaces
- Sobolev spaces
- Wave equations
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Small solutions to nonlinear wave equations in the Sobolev spaces. / Nakamura, M.; Ozawa, Tohru.
In: Houston Journal of Mathematics, Vol. 27, No. 3, 2001, p. 613-632.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Small solutions to nonlinear wave equations in the Sobolev spaces
AU - Nakamura, M.
AU - Ozawa, Tohru
PY - 2001
Y1 - 2001
N2 - The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space Ḣs = Ḣs(ℝn) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data (φ,ψ) ∈ Ḣ ≡ Ḣs ⊕ Ḣs-1, ∥(φ,ψ); Ḣ1/2∥ is relatively small with respect to ∥(φ,ψ); Ḣσ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u1+4/(n-1) near zero and has an arbitrary growth rate at infinity.
AB - The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space Ḣs = Ḣs(ℝn) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data (φ,ψ) ∈ Ḣ ≡ Ḣs ⊕ Ḣs-1, ∥(φ,ψ); Ḣ1/2∥ is relatively small with respect to ∥(φ,ψ); Ḣσ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u1+4/(n-1) near zero and has an arbitrary growth rate at infinity.
KW - Besov spaces
KW - Sobolev spaces
KW - Wave equations
UR - http://www.scopus.com/inward/record.url?scp=0035652960&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0035652960&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:0035652960
VL - 27
SP - 613
EP - 632
JO - Houston Journal of Mathematics
JF - Houston Journal of Mathematics
SN - 0362-1588
IS - 3
ER -