2017/05/15 by Xiangsheng Xu, Xu, Xiangsheng
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1705.05117
openalex publication_date 2017/05/15 · openalex created_date 2017/05/26 · openalex updated_date 2026/07/28
In this paper we study both the Cauchy problem and the initial boundary value problem for the equation ∂tu+div(∇Δu-\bf g(∇ u))=0. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vector-valued function \bf g is continuous and the initial datum u0 lies in C1(ℝN) with ∇ u0(x) being uniformly continuous and bounded on ℝN and that the global existence assertion also holds true if we assume that \bf g is locally Lipschitz and satisfies the growth condition |\bf g(ξ) |≤ c|ξ|α for some c>0, α∈ (2, 3), supℝN|∇ u0|