Existence and smoothing effects of the initial-boundary value problem for ∂u/∂t−Δσ(u) = 0 in time-dependent domains

We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form $ u_t − Δσ(u) = 0 $ in $ Q(0, T) $ with the initial and boundary conditions $ u(0) = u0 $ and $ u(t)|_{∂Ω(t)} = 0 $, where Ω(t) is a bounded domain in $ R^N $ for each t ≥ 0 and $ Q(0,T) = \bigcup_{0<t<T} \Omega(t) \times {t}, T>0 $. Our class of $ σ(u) $ includes $ σ(u) = |u|^m u $, $ σ(u) = u log(1 + |u|^m), 0 ≤ m ≤ 2, $ and $ σ(u) |u|^m u// \sqrt{1+|u|^2} $, $ 1 ≤ m ≤ 2 $ etc. We derive precise estimates for $ ||u(t)||_{Ω(t),∞} $ and $ ||∇σ(u(t))||_{Ω(t),2}^2, t > 0 $, depending on $ ||u_0||_{Ω(0),r} $ and the movement of $ ∂Ω(t) $.