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

Nakao Mitsuhiro

Opuscula Mathematica

2023

Vol. 43, no. 5

703--734

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 ut − 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 RN for each t ≥ 0 and [formula]. Our class of σ(u) includes σ(u) = |u|mu, σ(u) = u log(1 + |u|m), 0 ≤ m ≤ 2, and [formula]. We derive precise estimates for ∥u(t)∥Ω(t),∞ and ∥∇σ(u(t))∥2Ω(t),2, t > 0, depending on ∥u0∥Ω(0),r and the movement of ∂Ω(t).