Wellposedness results for a class of parabolic partial differential equations with hysteresis (Q733652)

In this paper the author studies the following equation in the unknown \(u\): \[ \frac{\partial}{\partial t}\left(u+\mathcal{F}(u)\right)+{\mathbf v}\cdot\nabla\left(u+\mathcal{F}(u)\right)-\Delta u=f, \] where \(\mathbf v\) is a given vector and \(\mathcal{F}\) is a hysteresis operator. She first gives an existence result for a variational solution in the case where \(\mathcal{F}\) is a Preisach operator, and then an uniqueness result if \(\mathcal{F}\) is Prandtl-Ishlinskii operator satisfying a suitable monotonicity property. Finally, the problem of continuous dependence on the data is also addressed.