00-83 Paleari, S., Bambusi, D., Cacciatori, S.

Exponential stability in a nonlinear string equation (40K, LaTeX) Feb 23, 00

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Abstract. We study the nonlinear wave equation $u_{tt}-c^2u_{xx}=\psi(u) \qquad u(0,t)=0=u(\pi,t)$ with an analytic nonlinearity of the type $\psi(u)=\pm u^3 + \sum_{k\ge 4}\alpha_k u^k$. On each small--energy surface we consider a solution of the linearized system with initial datum having the profile of an elliptic sinus: we show that solutions starting close to the corresponding phase space trajectory remain close to it for times growing exponentially with the inverse of the energy. To obtain the result we have to compute the resonant normal form of \ref{equa}, and we think this could be interesting in itself.

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