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Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

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Wu G., Yuan J.

Applicationes Mathematicae

2007

tom 34

nr 3

253-267

We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation $∂_{t}u - ∂³_{txx}u + 2κ∂_{x}u + ∂_{x}[g(u)/2] = γ(2∂_{x}u∂²_{xx}u + u∂³_{xxx}u)$ for the initial data u₀(x) in the Besov space $B^{s}_{p,r}(ℝ)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^{m}$-function (m ≥ 4) with g(0)=g'(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

Ograniczanie wyników

1 Applicationes Mathematicae

1 Wu G.

1 Yuan J.

1 2007

Wyniki wyszukiwania

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces