Solvability of nonlinear evolution equations with time-dependent constraints and applications.

*(English)*Zbl 0662.35054
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. BĂ©nilan, 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.

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. BĂ©nilan, 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 |