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

In this paper we study solvability of the Cauchy problem of the nonlinear beam equation ∂t2u+△2u=±up with initial data in Besov spaces. We prove that, for any 1≤q<∞1≤q<∞, the Cauchy problem of this equation is locally well-posed in the Besov spaces Ḃ2,qsp(Rn) and B2,qs(Rn), where sp=n2−4p−1 and s>sps>sp, and globally well-posed in these spaces if initial data are small. Moreover we obtain scattering results in Ḃ2,qsp(Rn).