Solvability of nonlinear evolution equations with time-dependent constraints and applications. (English) Zbl 0662.35054

Kenmochi, Nobuyuki

Bull. Fac. Educ., Chiba Univ., Part 2 30, 1-87 (1981).

A class of nonlinear parabolic partial differential equations in physics and mechanics can be interpreted as nonlinear evolution equations in Hilbert spaces of the form \[ (E)\quad du(t)/dt+\partial \phi^ t(u(t))\ni f(t),\quad 0<t<T, \] involving the subdifferential operators \(\partial \phi^ t\) of convex functions \(\phi^ t\). The purpose of the present paper is to study equation (E) from various view-points.
Throughout this paper we take an interest in the t-dependence of the mapping \(t\to \phi^ t\), which allows the effective domain \(D(\phi^ t)\) of \(\phi^ t\) to vary smoothly (but \(D(\phi^ t)\cap D(\phi^ s)=\phi\) may happen if \(t\neq s)\), and guarantees the solution of (E) in a certain sense. The t-dependence of \(t\to \phi^ t\) imposed in this paper is a modified version of that introduced by H. Attouch, Ph. Bénilan, A. Damlamian and C. Picard [C. R. Acad. Sci., Paris, Ser. I 279, 607-609 (1974; Zbl 0309.34049)], which is given by means of the regularizations \(\phi^ t_{\lambda}\) of \(\phi^ t.\)
In Chapter 1 the Cauchy problem \(CP(\phi^ t;f,u_ 0)\) for (E) with initial condition \(u(0)=u_ 0\) is formulated and the question of existence and uniqueness is considered under the above-mentioned t- dependence of \(t\to \phi^ t.\)
In Chapter 2, other aspects of equation (E) are considered; such as regularity and stability of solutions, characterizations of solutions in terms of variational inequalities, existence of periodic solutions, maximum and minimum periodic solutions, and convergence of solutions. In the final section of this chapter we consider \[ (E')\quad du(t)/dt+\partial \phi^ t(Bu(t))\ni f(t),\quad 0<t<T, \] where B is a given nonlinear operator; (E’) is a modified equation of (E) and is solved just as (E).
In Chapter 3 we give some applications of abstract results obtained in Chapters 1 and 2 to a class of nonlinear parabolic partial differential equations with time-dependent constraints; for example, quasi-linear heat equations with obstacles, equations of the Stokes-type in non-cylindrical domains and free boundary problems of the Stefan-type for nonlinear parabolic equations.

MSC:

35K55	Nonlinear parabolic equations
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35K15	Initial value problems for second-order parabolic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs
35B10	Periodic solutions to PDEs
35B35	Stability in context of PDEs
35K85	Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
80A20	Heat and mass transfer, heat flow (MSC2010)
35R35	Free boundary problems for PDEs

Keywords:

nonlinear evolution equations; Hilbert spaces; subdifferential operators; convex functions; regularizations; Cauchy problem; existence; uniqueness; regularity; stability; characterizations of solutions; variational inequalities; periodic solutions; convergence of solutions; quasi-linear; obstacles; equations of the Stokes-type; free boundary problems of the Stefan-type

Citations:

Zbl 0309.34049