VISCOSITY SOLUTION TO A DIRICHLET INITIAL-BOUNDARY VALUE PROBLEM OF A MODEL FOR DISLOCATION MOTION

. In this article, we shall prove the existence and uniqueness of viscosity solution to an initial (Dirichlet)-boundary value problem for a continuum phase-field model, which is a quasi-linear, non-uniform, degenerate parabolic equation of second order with a non-smooth, super-linear gradient term. This model is proposed by Acharya et al., to describe the motion of dislocations that is a kind of one-dimensional defect that significantly impact the properties of crystalline materials, such as conductivity and plasticity.