On a class of fully nonlinear parabolic equations

We study the homogeneous Dirichlet problem for the fully nonlinear equation u(t) = vertical bar Delta u vertical bar(m- 2) Delta u - d vertical bar u vertical bar(sigma-2) u + f in Q(T) = Omega x (0, T), with the parameters m > 1, sigma > 1 and d >= 0. At the points where Delta u = 0, the equation degenerates if m > 2, or becomes singular if m is an element of (1, 2). We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as t -> infinity. Sufficient conditions for exponential or power decay of parallel to del u(t)parallel to(2),(Omega) are derived. It is proved that for certain ranges of the exponents m and s, every strong solution vanishes in a finite time.