Existence of solutions and separation from singularities for a class of\n fourth order degenerate parabolic equations

2010/09/16 by Giulio Schimperna, Schimperna, Giulio, Sergey Zelik +1 · 1 citation
Computer Science · Engineering · #35K35 #35K65 #37L30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1009.3148

openalex publication_date 2010/09/16 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

A nonlinear parabolic equation of the fourth order is analyzed. The equation\nis characterized by a mobility coefficient that degenerates at 0. Existence of\nat least one weak solution is proved by using a regularization procedure and\ndeducing suitable a-priori estimates. If a viscosity term is added and\nadditional conditions on the nonlinear terms are assumed, then it is proved\nthat any weak solution becomes instantaneously strictly positive. This in\nparticular implies uniqueness for strictly positive times and further\ntime-regularization properties. The long-time behavior of the problem is also\ninvestigated and the existence of trajectory attractors and, under more\nrestrictive conditions, of strong global attractors is shown.\n

Citations

• Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility

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• On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

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