Abstract :
This paper is devoted to studying the existence and asymptotic behavior of solutions to
a nonlinear parabolic equation of fourth order: ut +∇·(|∇ u|p−2∇ u) = f (u) in Ω ⊂ RN
with boundary condition u = u = 0 and initial data u0. The substantial difficulty is
that the general maximum principle does not hold for it. The solutions are obtained for
both the steady-state case and the developing case by the fixed point theorem and the
semi-discretization method. Unlike the general procedures used in the previous papers
on the subject, we introduce two families of approximate solutions with determining the
uniform bounds of derivatives with respect to the time and space variables, respectively. By
a compactness argument with necessary estimates, we show that the two approximation
sequences converge to the same limit, i.e., the solution to be determined. In addition,
the decays of solutions towards the constant steady states are established via the entropy
method. Finally, it is interesting to observe that the solutions just tend to the initial data u0
as p→∞.
