  In this paper we establish three global in time results for two fourth order
nonlinear parabolic equations. The first of such equations involves the Hessian
and appears in epitaxial growth. For such equation we give conditions ensuring
the global existence of solution. For certain regime of the parameters, our
size condition involves the norm in a critical space with respect to the
scaling of the equation and improves previous existing results in the
literature for this equation. The second of the equations under study is a thin
film equation with a porous medium nonlinearity. For this equation we establish
conditions leading to the global existence of solution.

Analysis of PDEs