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A class of nonlinear elliptic-parabolic equations with time-dependent constraints. (English) Zbl 0635.35043
This 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.

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)
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