Radius of analyticity for the Camassa–Holm equation on the line. (September 2018) - Himonas, A. Alexandrou; Petronilho, Gerson
Abstract: Using estimates in Sobolev spaces we prove that the solution to the Cauchy problem for the Camassa–Holm equation on the line with analytic initial data u 0 ( x ) and satisfying the McKean condition, that is the quantity m 0 ( x ) = ( 1 − ∂ x 2 ) u 0 ( x ) does not change sign, is analytic in the spatial variable for all time. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity r ( t ) given by r ( t ) ≥ A − 1 ( 1 + C 1 B t ) − 1 exp − C 0 ‖ u 0 ‖ H 1 t , where A, B, C 1 and C 0 are suitable positive constants.