Analyticity and smoothing effect for the fifth order KdV type equation

作者：Tomoeda Kyoko^*

来源：Proceedings of the Japan Academy Series A-Mathematical Sciences, 2010, 86(7): 101-106.

DOI：10.3792/pjaa.86.101

• Analytic
• smoothing effect
• fifth order KdV equation
• KdV hierarchy

We consider the initial value problem for the reduced fifth order KdV type equation: partial derivative(t)u - partial derivative(5)(x)u - 10 partial derivative(x)(u(3)) + 5 partial derivative(x)(partial derivative(x)u)(2) = 0 which is obtained by removing the nonlinear term 10 partial derivative(x)(u partial derivative(2)(x)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 phi is an element of H-s(R) (s > 1/8) satisfies the condition
Sigma(infinity)(k=0) A(0)(k)/k !parallel to(x partial derivative(x))(k) phi parallel to(Hs) < infinity,
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 [5].

• 出版日期2010-7