A class of nonlinear elliptic-parabolic equations with time-dependent constraints.

*(English)*Zbl 0635.35043This paper is devoted to the study of nonlinear evolution equations of the form
\[
(0.1)\quad (d/dt)u(t)+\partial \phi^ t(v(t))\ni f(t),\quad v(t)\in Bu(t),\quad 0<t<T,
\]
in a Hilbert space H, where \(\partial \phi^ t\) is the subdifferential of a convex function \(\phi^ t\) defined on H, B is a maximal monotone operator in H. The Cauchy problem \(CP(\phi^ t,B,f,u_ 0)\) for (0.1), with the initial condition \(u(0)=u_ 0\) is considered. The existence and uniqueness questions are analyzed. A priori estimates for the solution, uniform with respect to \(\phi^ t\), B, f, \(u_ 0\) are established and a continuous dependence of the solution upon the data is shown.

In the case of a bi-Lipschitz subdifferential operator B on H, the problem \(CP(\phi^ t,B,f,u_ 0)\) was studied by the first author [Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ. 30, 1-87 (1981)]. Exploiting arguments developed by the first author [Nonlinear Anal., Theory Methods Appl. 5, 71-80 (1981; Zbl 0458.35053)], we extend results of the first paper to the case of maximal monotone operators B.

The present paper is motivated by a study of elliptic-parabolic variational inequalities with time-dependent obstacles which arise, in particular, from modelling processes with free boundaries such as multi- phase flows through porous media and processes of electrochemical technology.

In the case of a bi-Lipschitz subdifferential operator B on H, the problem \(CP(\phi^ t,B,f,u_ 0)\) was studied by the first author [Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ. 30, 1-87 (1981)]. Exploiting arguments developed by the first author [Nonlinear Anal., Theory Methods Appl. 5, 71-80 (1981; Zbl 0458.35053)], we extend results of the first paper to the case of maximal monotone operators B.

The present paper is motivated by a study of elliptic-parabolic variational inequalities with time-dependent obstacles which arise, in particular, from modelling processes with free boundaries such as multi- phase flows through porous media and processes of electrochemical technology.

##### MSC:

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

35B45 | A priori estimates in context of PDEs |

35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |

47H05 | Monotone operators and generalizations |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

##### Keywords:

nonlinear evolution equations; subdifferential; convex function; maximal monotone operator; Cauchy problem; existence; uniqueness; A priori estimates; continuous dependence; elliptic-parabolic variational inequalities; time-dependent obstacles; multi-phase flows; porous media; electrochemical technology
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\textit{N. Kenmochi} and \textit{I. Pawlow}, Nonlinear Anal., Theory Methods Appl. 10, 1181--1202 (1986; Zbl 0635.35043)

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##### References:

[1] | Alt, H.W.; Luckhaus, S., Quasilinear elliptic – parabolic differential equations, Math. Z., 183, 311-341, (1983) · Zbl 0497.35049 |

[2] | Alt, H.W.; Luckhaus, S.; Visintin, A., On nonstationary flow through porous media, Annali mat. pura appl., 136, 303-316, (1984) · Zbl 0552.76075 |

[3] | Carroll, R.W.; Showalter, R.E., Singular and degenerate Cauchy problems, () |

[4] | DiBenedetto, E.; Showalter, R.E., Implicit degenerate evolution equations and applications, SIAM J. math. analysis, 12, 731-751, (1981) · Zbl 0477.47037 |

[5] | DiBenedetto, E.; Gariepy, R., Local behaviour of solutions of an elliptic-parabolic equation, (), MRC · Zbl 0627.35052 |

[6] | Elliott, C.M., A variational inequality formulation of a steady state electrochemical machining free boundary problem, (), 505-512 |

[7] | Elliott, C.M.; Ockendon, J.R., Weak and variational methods for moving boundary problems, () · Zbl 0476.35080 |

[8] | Fasano, A.; Primicerio, M., Partially saturated porous media, J. inst. math. applic., 23, 503-517, (1979) · Zbl 0428.76076 |

[9] | Grange, O.; Mignot, F., Sur la résolution d’une équation et d’une inéquation paraboliques d’évolution, J. funct. analysis, 11, 77-92, (1972) · Zbl 0251.35055 |

[10] | Hornung, U., A parabolic – elliptic variational inequality, Manuscripta math., 39, 155-172, (1982) · Zbl 0502.35055 |

[11] | Hornung, U., A unilateral boundary value problem for unsteady waterflow in porous media, (), 59-94 |

[12] | Kenmochi, N., Méthode de compacité et résolution de problèmes variationnels paraboliques quasi-linéaires ou semilinéaires, () |

[13] | Kenmochi, N., On the quasi-linear heat equation with time-dependent obstacles, Nonlinear analysis, 5, 71-80, (1981) · Zbl 0458.35053 |

[14] | Kenmochi, N., Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. fac. education, chiba univ., 30, 1-87, (1981), (Part II) |

[15] | K\scENMOCHI N. & P\scAWLOW I., Elliptic—parabolic variational inequalities with time dependent constraints arising from free boundary problems, in preparation. |

[16] | Kröner, D., Parabolic regularization and behaviour of the free boundary for unsaturated flow in a porous medium, J. reine angew. math., 348, 180-196, (1984) · Zbl 0517.76095 |

[17] | Kröner, D.; Rodriguez, J.-F., Global behaviour for bounded solutions of a porous media equation of elliptic—parabolic type, (), SFB 72 |

[18] | Kuttler, K.L., Degenerate variational inequalities of evolution, Nonlinear analysis, 8, 837-850, (1984) · Zbl 0549.49004 |

[19] | McGeough, J.A., Free and moving boundary problems in electrochemical machining and flame fronts, (), 472-482 · Zbl 0501.35077 |

[20] | Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Adv. math., 3, 510-585, (1969) · Zbl 0192.49101 |

[21] | Watanabe, J., Approximation of nonlinear problems of a certain type, Numerical analysis of evolution equations, Lecture notes in numer. appl. analysis, 1, 147-163, (1979) |

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