Solvability of the Cauchy problem of nonlinear beam equations in Besov spaces

Abstract

In this paper we study solvability of the Cauchy problem of the nonlinear beam equation ∂ t 2 u + △ 2 u = ± u p with initial data in Besov spaces. We prove that, for any 1 ≤ q < ∞ , the Cauchy problem of this equation is locally well-posed in the Besov spaces B ̇ 2 , q s p ( R n ) and B 2 , q s ( R n ) , where s p = n 2 − 4 p − 1 and s > s p , and globally well-posed in these spaces if initial data are small. Moreover we obtain scattering results in B ̇ 2 , q s p ( R n ) .