ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A HIGHER-ORDER KDV-TYPE EQUATION WITH CRITICAL NONLINEARITY

We consider the Cauchy problem of the higher-order KdV-type equation: partial derivative(t)u + 1/m vertical bar partial derivative(x)vertical bar(m-1)partial derivative(x)u = partial derivative(x)(u(m)) where m >= 4. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.