2015/06/18 by Seonghak Kim, Kim, Seonghak, Baisheng Yan +1
Computer Science · Mathematics · #35D30 #35K20 #35M13 #49K20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1506.05847
openalex publication_date 2015/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona-Malik model in image processing and the other that of Höllig's model related to the Clausius-Duhem inequality in the second law of thermodynamics.