We consider the Cauchy problem for the Kirchhoff equation on T^d with initial data of small amplitude ε in Sobolev class. We prove a lower bound ε^−4 for the existence time, which improves the bound ε^−2 given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.
On the existence time for the Kirchhoff equation with periodic boundary conditions / Baldi, Pietro; Haus, Emanuele. - In: NONLINEARITY. - ISSN 0951-7715. - 33:1(2020), pp. 196-223. [10.1088/1361-6544/ab4c7b]
On the existence time for the Kirchhoff equation with periodic boundary conditions
Pietro Baldi;
2020
Abstract
We consider the Cauchy problem for the Kirchhoff equation on T^d with initial data of small amplitude ε in Sobolev class. We prove a lower bound ε^−4 for the existence time, which improves the bound ε^−2 given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
