### Nonlinear systems of parabolic PDE's for phase change problem

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In this paper a model for the recovery of human and economic activities in a region, which underwent a serious disaster, is proposed. The model treats the case that the disaster region has an industrial collaboration with a non-disaster region in the production system and, especially, depends upon each other in technological development. The economic growth model is based on the classical theory of R. M. Solow (1956), and the full model is described as a nonlinear system of ordinary differential...

Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize...

We consider an abstract formulation for a class of parabolic quasi-variational inequalities or quasi-linear PDEs, which are generated by subdifferentials of convex functions with various nonlocal constraints depending on the unknown functions. In this paper we specify a class of convex functions ${\phi}^{t}(v;\xb7)$ on a real Hilbert space H, with parameters 0 ≤ t ≤ T and v in a set of functions from [-δ₀,T], 0 < δ₀ < ∞, into H, in order to formulate an evolution equation of the form ${u}^{\text{'}}\left(t\right)+\partial {\phi}^{t}(u;u\left(t\right))\ni f\left(t\right)$, 0 < t < T, in H. Our...

We consider a phase-field model of grain structure evolution, which appears in materials sciences. In this paper we study the grain boundary motion model of Kobayashi-Warren-Carter type, which contains a singular diffusivity. The main objective of this paper is to show the existence of solutions in a generalized sense. Moreover, we show the uniqueness of solutions for the model in one-dimensional space.

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