Threshold phenomenon for homogenized fronts in random elastic media

We consider a model for the motion of phase interface in an elastic medium, example, twin boundary martensite. The is given by semilinear parabolic equation with fractional Laplacian as regularizing operator, stemming from interaction front its environment. We show that presence randomly distributed, localized obstacles leads to threshold phenomenon, i.e., stationary solutions exist up positive, critical driving force leading stick-slip behaviour boundary. main result proved explicit construction viscosity supersolution evolution and based on percolation obstacle sites. Furthermore, we derive homogenization such fronts case half-Laplacian pinning regime.