On The Initial Boundary Value Problem Of A Nonlinear Fourth-order Parabolic Equation

In this paper , we study the nonlinear fourth - order parabolic equation :This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling . Here we study the initial boundary value problem:The existence of nonnegative weak solutions globally in time of this equation in one space dimension is shown . The proof is based on an exponential transformation of variables and entropy estimates . In particular , when f(t) = 0 , we prove exponential decay of logn(x, t) towards the constant steady state log n_∞ = 0 in the L~1 norm for long times and give the explicit rate of decay. Finally , we prove the existence of several families of Lyapunov functionals .