About a Singular Parabolic Equation Arising in Thin Film Dynamics and in the Ricci Flow for Complete IR2

Let u(x,t) be the thickness at the location x ≡ (x ₁, x ₂) ∈ R² and at time t ∈ R⁺ , of a thin (100Å–1000Å) colloidal film spread over a flat surface. According to [1,9], if the van der Waals forces are repulsive, a model for the evolution of u is the Cauchy problem, 1.1 u t − Δ ln u = 0   in   S T ≡ I R 2 × ( 0 , T ) ,   a n d   u ( ⋅ , 0 ) = u o , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq715.tif"/>