Existence and long time behaviour of solutions to obstacle thermistor equations
Walter Allegretto Yanping Lin Shuqing Ma
In this paper we introduce an obstacle thermistor system. The existence of weak solutions to the steady-state systems and capacity solutions to the time dependent systems are obtained by a penalized method under reasonable assumptions for the initial and boundary data. At the same time, we prove that there exists a uniform absorbing set for nonnegative initial data in $L_2(\Omega)$. Finally for smooth initial data a global attractor to the system is obtained by a series of Campanato space arguments.
keywords: capacity solutions Obstacle thermistor equations global attractors.
Hölder continuous solutions of an obstacle thermistor problem
Walter Allegretto Yanping Lin Shuqing Ma
In this paper we consider a thermistor problem with a current source, i.e., a nonlocal boundary condition. The electric potential is unknown on part of the boundary, but the current through it is known. We apply a decomposition technique and transform the equation satisfied by the potential into two elliptic problems with usual boundary conditions. The unique solvability of the initial boundary value problem is achieved.
keywords: existence uniqueness. current driven Obstacle thermistor problem
On the box method for a non-local parabolic variational inequality
Walter Allegretto Yanping Lin Shuqing Ma
In this paper we study a box scheme (or finite volume element method) for a non-local nonlinear parabolic variational inequality arising in the study of thermistor problems. Under some assumptions on the data and regularity of the solution, optimal error estimates in the $H^1$-norm are attained.
keywords: Box methods non-local thermistor and variational inequality.

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