Smoothing and perturbation for some fourth order linear parabolic equations in

We solve some fourth order parabolic equations, obtained from perturbations of the parabolic bi-Laplacian equation, with special focus on smoothing estimates. Several classes of initial data are considered including data in Lebesgue and Bessel–Lebesgue spaces. Robustness and convergence with respect to the perturbation are also obtained. Also, initial data in large uniform Bessel–Lebesgue spaces are considered as well as equations with higher order powers of the Laplacian.