Abstract In this paper we consider some fourth order linear and semilinear equations in R N and make a detailed study of the solvability of the Cauchy problem. For the linear equation we consider some weakly integrable potential terms, and for any 1 p ∞ prove that for a suitable family of Bessel potential spaces, H p α ( R N ) , the linear equation defines a strongly continuous analytic semigroup. Using this result, we prove that the nonlinear problems we consider can be solved for initial data in L p ( R N ) and in H p 2 ( R N ) . We also find the corresponding critical exponents, that is, the largest growth allowed for the nonlinear terms for these classes of initial data.