Abstract:

In this paper， we study the Cauchy problem for the 4-th order nonlinear equation {\partial }_{t}h+{\partial }_x^{\mathrm{4}}h+{\partial }_x^{\mathrm{2}}\left({\left|{\partial }_{x}h\right|}^{\alpha }\right)=\mathrm{0} in one dimension for the initial data h_{\mathrm{0}} where \alpha \ge \mathrm{5} and \alpha \in \mathbb{R}. Making use of some subtle estimates of the corresponding linear equation and the nonlinear term， Littlewood-Paley theory， two-norm method， and contraction mapping principle， we obtain the local well-posedness result in nonhomogeneous Besov spaces.

Key words: surface growth model, Cauchy problem, well-posedness, Besov spaces, Littlewood-Paley theory