We study the homogeneous Dirichlet problem for the fully nonlinear equation u t = Δ u m - 2 Δ u - d u σ - 2 u + f in Q T = Ω × ( 0 , T ) , u_{t}=Delta u^{m-2}Delta u-du^{sigma-2}u+fquadtext{in ${Q_{T}=Omega times(0,T)}$,} with the parameters m 1 {m1} , σ 1 {sigma1} and d ≥ 0 {dgeq 0} . At the points where Δ u = 0 {Delta u=0} , the equation degenerates if m 2 {m2} , or becomes singular if m ∈ ( 1 , 2 ) {min(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as t → ∞ {ttoinfty} . Sufficient conditions for exponential or power decay of ∥ ∇ u ( t ) ∥ 2 , Ω {nabla u(t)_{2,Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.
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