On Lipschitz solutions for some forward–backward parabolic equations

Kim, Seonghak; Yan, Baisheng

doi:10.1016/j.anihpc.2017.03.001

Kim, Seonghak ; Yan, Baisheng

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 65-100.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2017.03.001

Classification : 35M13, 35K20, 35D30, 49K20
Mots clés : Forward–backward parabolic equations, Partial differential inclusions, Convex integration, Baire's category method, Infinitely many Lipschitz solutions

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     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},
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     year = {2018},
     doi = {10.1016/j.anihpc.2017.03.001},
     mrnumber = {3739928},
     zbl = {1378.35153},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.001/}
}

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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. http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.001/

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