Nobuyuki Kenmochi

Parabolic PDEs with hysteresis and quasivariational inequalities

w0 being prescribed as an initial output of w; its input–output relation is illustrated in Fig. 1.1 (see [1, Part I, III-2] for the precise de nition); in the gure fa and fd are continuous and nondecreasing functions on R such that fa≤fd on R. We refer for some results on this sort of model to [1–4]. As is well known [1, Part I, III-2], (1.2) is equivalent to the following variational inequality wt(x; t) + @Iu(x; t)(w(x; t))3 0; (x; t)∈QT ; (1.3)

Viscosity approach to modelling non-isothermal diffusive phase separation

A nonlinear parabolic system with a singular evolution term, arising from modelling dynamic phenomena of the non-isothermal diffusive phase separation, is studied. The system is subject to constraints entering into the main part of one of the equations. In the paper, questions related to the existence and uniqueness of solutions to the singular problem are studied with the use of viscosity approximations. To this purpose, a special regularization technique has been applied to the singular evolution term.

Parabolic-elliptic free boundary problems with time-dependent obstacles

In the paper, parabolic-elliptic problems with time-dependent obstacles and variable boundary conditions are considered. The obstacles are either concentrated at a given time-dependent part of the boundary of a geometric domain or within a prescribed time-dependent subdomain. The problems are fourmulated as a Cauchy problem in Hilbert space. Existence and uniqueness results are established, in particular refering to some models of flows in porous media and electrochemical machining processes with moving control actions.

Asymptotic behavior of solutions to a multi-phase Stefan problem

This paper is concerned with the asymptotic behavior of weak solutions to a multi-phase Stefan problem for a quasi-linear heat equation of the form ρ(v)t - Δv=ƒ in several space variables, with Dirichlet-Neumann boundary condition on the fixed boundary. We shall discuss the asymptotic convergence of the enthalpy and temperature inL2(Ω) andH1(Ω), respectively, when the prescribed boundary data asymptotically converge in some sense. Our approach to the investigation of the asymptotic convergence of solutions is based on the theory of nonlinear evolution equations governed by time-dependent subdifferential operators in Hilbert spaces. The results obtained in this paper improve on those established so far.

A New Proof of the Uniqueness of Solutions to Two-Phase Stefan Problems for Nonlinear Parabolic Equations

Let us consider a two-phase Stefan problem described as follows: Find a function,u=u(t,x) on Q=(O,T)×(0,1), O<T<∞, and a curve x = l(t), 0<l<1, on [O,T] such that

Asymptotic behaviour of solutions to parabolic-elliptic variational inequalities

\begin{array}{*{20}{c}} {\rho {{{(u)}}_{t}} - a{{{({{u}_{x}})}}_{x}} + h(t,x) = \left( {\begin{array}{*{20}{c}} {{{f}_{o}}(t,x)} & {in Q_{\ell }^{o},} \\ {{{f}_{l}}(t,x)} & {in Q_{\ell }^{l},} \\ \end{array} } \right.} \\ {\begin{array}{*{20}{c}} {h(t,x) \in g(t,x,u(t,x))} & {for a.e. (t,x) \in Q,} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {Q_{\ell }^{o} = \{ (t,x); 0 < t < T,} & {0 < x < \ell (t)\} ,} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {Q_{\ell }^{l} = \{ (t,x); 0 < t < T,} & {\ell (t) < x < l\} ,} \\ \end{array} } \\ \end{array}

Asymptotic Stability of Solutions to a Two-Phase Stefan Problem with Nonlinear Boundary Condition of Signorini Type

(0.1)