We consider a nonlinear diffusion equation with irreversible property and construct a unique strong solution by using implicit time discretization. A new regularity estimate for the classical obstacle problem is established and is used in the construction of the strong solution.

As an application, we consider a quasi-static fracture model of brittle material using the idea of the phase field model. The Francfort-Marigo energy which is based on the classical Griffith theory is introduced, where the sharp crack profile is approximated by a smooth damage function using the idea of the Ambrosio-Tortorelli regularization. The crack propagation model is derived as a gradient flow of the energy of the damage variable with an irreversible constraint. Some numerical
examples in various settings computed by finite element method are also presented in the talk.

The contents is based on the joint works with Goro Akagi (Kobe Univ.) and with Takeshi Takaishi (Hiroshima Kokusai Gakuin Univ.).

This lecture will be given in Japanese.