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Solvability of a class of phase field systems related to a sliding mode control problem. (English) Zbl 1413.35265
Summary: We consider a phase-field system of Caginalp type perturbed by the presence of an additional maximal monotone nonlinearity. Such a system arises from a recent study of a sliding mode control problem. We prove the existence of strong solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution.

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35K25 Higher-order parabolic equations
35B25 Singular perturbations in context of PDEs
35D30 Weak solutions to PDEs
80A22 Stefan problems, phase changes, etc.
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[1] V. Barbu: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics, Springer, New York, 2010. · Zbl 1197.35002
[2] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, E. Rocca: Sliding mode control for a nonlinear phase-field system. Preprint arXiv: 1506.01665[math. AP] (2015), 1–28.
[3] H. Brézis: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies 5. Notas de Matemática (50), North-Holland Publishing, Amsterdam-London; American Elsevier Publishing, New York, 1973. · Zbl 0252.47055
[4] M. Brokate, J. Sprekels: Hysteresis and Phase Transitions. Applied Mathematical Sciences 121, Springer, New York, 1996. · Zbl 0951.74002
[5] G. Caginalp: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92 (1986), 205–245. · Zbl 0608.35080 · doi:10.1007/BF00254827
[6] P. Colli, G. Gilardi, G. Marinoschi: A boundary control problem for a possibly singular phase field system with dynamic boundary conditions. J. Math. Anal. Appl. 434 (2016), 432–463. · Zbl 1327.49007 · doi:10.1016/j.jmaa.2015.09.011
[7] P. Colli, G. Gilardi, G. Marinoschi, E. Rocca: Optimal control for a phase field system with a possibly singular potential. Math. Control Relat. Fields 6 (2016), 95–112. · Zbl 1335.49008 · doi:10.3934/mcrf.2016.6.95
[8] P. Colli, G. Marinoschi, E. Rocca: Sharp interface control in a Penrose-Fife model. ESAIM, Control Optim. Calc. Var. 22 (2016), 473–499. · Zbl 1338.49007 · doi:10.1051/cocv/2015014
[9] A. Damlamian: Some results on the multi-phase Stefan problem. Commun. Partial Differ. Equations 2 (1977), 1017–1044. · Zbl 0399.35054 · doi:10.1080/03605307708820053
[10] E. DiBenedetto: Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J. 32 (1983), 83–118. · Zbl 0526.35042 · doi:10.1512/iumj.1983.32.32008
[11] G. Duvaut: Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré). C. R. Acad. Sci., Paris, Sér. A 276 (1973), 1461–1463. (In French.) · Zbl 0258.35037
[12] C. M. Elliott, S. Zheng: Global existence and stability of solutions to the phase field equations. Free Boundary Value Problems. Proc. Conf. Oberwolfach, 1989, Internat. Ser. Numer. Math. 95, Birkhäuser, Basel, 1990, pp. 46–58.
[13] A. Friedman: The Stefan problem in several space variables. Trans. Am. Math. Soc. 133 (1968), 51–87. · Zbl 0162.41903 · doi:10.1090/S0002-9947-1968-0227625-7
[14] M. Grasselli, H. Petzeltová, G. Schimperna: Long time behavior of solutions to the Caginalp system with singular potential. Z. Anal. Anwend. 25 (2006), 51–72. · Zbl 1128.35021 · doi:10.4171/ZAA/1277
[15] K.-H. Hoffmann, L. S. Jiang: Optimal control of a phase field model for solidification. Numer. Funct. Anal. Optimization 13 (1992), 11–27. · Zbl 0724.49003 · doi:10.1080/01630569208816458
[16] K.-H. Hoffmann, N. Kenmochi, M. Kubo, N. Yamazaki: Optimal control problems for models of phase-field type with hysteresis of play operator. Adv. Math. Sci. Appl. 17 (2007), 305–336. · Zbl 1287.49005
[17] K. M. Hui: Existence of solutions of the very fast diffusion equation in bounded and unbounded domain. Math. Ann. 339 (2007), 395–443. · Zbl 1145.35075 · doi:10.1007/s00208-007-0119-x
[18] N. Kenmochi, M. Niezgódka: Evolution systems of nonlinear variational inequalities arising from phase change problems. Nonlinear Anal., Theory Methods Appl. 22 (1994), 1163–1180. · Zbl 0827.35070 · doi:10.1016/0362-546X(94)90235-6
[19] Ph. Laurençot: Long-time behaviour for a model of phase-field type. Proc. R. Soc. Edinb., Sect. A 126 (1996), 167–185. · Zbl 0851.35055 · doi:10.1017/S0308210500030663
[20] R. E. Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49, Amer. Math. Soc., Providence, 1997. · Zbl 0870.35004
[21] J. Simon: Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl., IV. Ser. 146 (1987), 65–96. · Zbl 0629.46031 · doi:10.1007/BF01762360
[22] J. L. Vázquez: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
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