Positive solutions to a nonlinear fourth-order partial differential equation

A class of fourth-order partial differential equations with Dirichlet boundary conditions which has important applications in phase transformation theory and in the description of the motion of a very thin layer of viscous incompressible fluid is studied. By transforming the higher order problem into a second-order elliptic system and using a fixed point method, the existence and uniqueness of steady state solutions are obtained. Moreover, by using a semi-discrete method, the existence of strong solutions of the evolution equation is obtained if the initial function is in the neighborhood of a positive steady state solution. Finally, the solution of the evolution equation converges to its steady state solution as the time t→+∞t→+∞.