Higher order nonlinear degenerate parabolic equations

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

“…x,t (Ω T ), see (1.4), as proved in §7 of [8]. Moreover, the above extends their existence and regularity results to higher space dimensions.…”

“…Hence we conclude from (3.59a-c) that (1.3) holds with {|u| > 0} replaced by {u > 0}. This is the weak formulation of (1.1a) introduced by [8] in one space dimension. A weak formulation of the boundary condition b(u) ∂Δ 2 u ∂ν = 0 is also incorporated in (1.3).…”

“…Existence of Hölder continuous nonnegative solutions to (P) in one space dimension (d = 1) has been established in [8]. They used a very weak solution concept, which basically says that a function u solves (P) if for all η ∈ L 2 (0, T ; H 1 (Ω)) x,t (Ω T ).…”

“…In §3 we establish convergence of our approximation. Unlike the numerical approximations of degenerate fourth order problems, see [4,5,12,6,3], where convergence is only established in one space dimension; we are able, by exploiting the fact that the operator is of higher order, to show convergence in all space dimensions (1 ≤ d ≤ 3) to a solution u satisfying the solution concept (1.3) of [8]. This, in particular, extends the existence and regularity results of [8] from one space dimension to higher space dimensions.…”

Point Rupture Solutions of Singular Elliptic Equations in N-D

“…When n = 1, m = 1, it describes a thin jet in a Hele-Shaw cell [1], [5], [8], [9]; when n = m = 3 it describes fluid droplets hanging from a ceiling [10]; when n = 0 and m = 1, it describes solidification of a hyper-cooled melt [3], [4]; and when n = 3, m = −1, it models van der Waals force driven thin film [7], [12], [18], [19], [20], when the space dimension is one R. Laugesen and M. Pugh [16] studied rigorously, in a general setting, positive periodic steady states and touchdown steady states solution. F. Bernis and A. Friedman in [2] established the existence of weak solutions and showed that the support of the thin film will expand with time. Equation (1.3) models the dynamic of thin films equation, using the pressure as defined earlier with Neumann boundary condition ∂u ∂n = 0 on ∂Ω.…”