Well-posedness and Ill-posedness for the Nonlinear Beam Equation

Abstract

We investigate Strichartz estimates for the nonlinear beam equation with initial data $f\in {\dot{H}}^s,g\in {\dot{H}}^{s-2}$ and $f\in H^s,g\in H^{s-2}$. We extend results of H. Lindblad and C. D.Sogge [10] and T. Cazenave and F. B. Weissler [4] to nonlinear beam equations to determine the minimal regularity that is needed to prove well-posedness and scattering results with low regularity data. Finally, we also use small dispersion analysis of M. Christ, J. Colliander and T. Tao [2] to prove the nonlinear beam equation is ill-posed in defocusing case $\omega =-1$ when $0<s<s_c=\frac{n}{2}-\frac{4}{\kappa -1}$.

arXiv admin note: text overlap with arXiv:math/0311048 by other authors

References in corpus (1)

• Ill-posedness for nonlinear Schrodinger and wave equations