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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.
Keywords: Forward–backward parabolic equations, Partial differential inclusions, Convex integration, Baire's category method, Infinitely many Lipschitz solutions
@article{AIHPC_2018__35_1_65_0,
author = {Kim, Seonghak and Yan, Baisheng},
title = {On {Lipschitz} solutions for some forward{\textendash}backward parabolic equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {65--100},
publisher = {Elsevier},
volume = {35},
number = {1},
year = {2018},
doi = {10.1016/j.anihpc.2017.03.001},
mrnumber = {3739928},
zbl = {1378.35153},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.03.001/}
}
TY - JOUR AU - Kim, Seonghak AU - Yan, Baisheng TI - On Lipschitz solutions for some forward–backward parabolic equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 65 EP - 100 VL - 35 IS - 1 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.03.001/ DO - 10.1016/j.anihpc.2017.03.001 LA - en ID - AIHPC_2018__35_1_65_0 ER -
%0 Journal Article %A Kim, Seonghak %A Yan, Baisheng %T On Lipschitz solutions for some forward–backward parabolic equations %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 65-100 %V 35 %N 1 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.03.001/ %R 10.1016/j.anihpc.2017.03.001 %G en %F AIHPC_2018__35_1_65_0
Kim, Seonghak; Yan, Baisheng. On Lipschitz solutions for some forward–backward parabolic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 65-100. doi: 10.1016/j.anihpc.2017.03.001
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