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Analyticity and smoothing effect for the fifth order KdV type equation. (English) Zbl 1205.35276

Summary: We consider the initial value problem for the reduced fifth-order KdV type equation: \(\partial_tu-\partial_x^5 u-10\partial_x(u^3)+ 5\partial_x(\partial_xu)^2=0\) which is obtained by removing the nonlinear term \(10\partial_x(u\partial_x^2 u)\) from the fifth order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data \(\varphi\in H^{s}(\mathbb R)\) \((s>1/8)\) satisfies the condition
\[ \sum_{k=0}^\infty \frac{A_0^k}{k!} \big\|(x\partial_x)^k \varphi\big\|_{H^s}<\infty \]
for some constant \(A_0\) \((0<A_0<1)\). Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the argument used in [K. Kato and T. Ogawa, Math. Ann. 316, No. 3, 577–608 (2000; Zbl 0956.35115)].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0956.35115
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References:

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