On Lipschitz solutions for some forward–backward parabolic equations
Seonghak Kim
Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, PR ChinaBaisheng Yan
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
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Abstract
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. Under this framework, 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.
Cite this article
Seonghak Kim, Baisheng Yan, On Lipschitz solutions for some forward–backward parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 1, pp. 65–100
DOI 10.1016/J.ANIHPC.2017.03.001