Blow-up of solutions of a nonlinear parabolic equation in damage mechanics

A fully nonlinear, degenerate parabolic equation arising in the theory of damage mechanics is shown to be well-posed. Its solutions blow up in finite time and, under suitable conditions on the initial configuration, the blow-up set, corresponding to the portion of the material which breaks at the blow-up time, is an interval of nonzero measure. In a special but physically relevant case the problem reduces to the study of the blow-up set of solutions of the quasilinear equation u(t) = u(alpha)(lambda(2)u(xx) + u) with homogeneous Neumann boundary data, and the size of the blow-up set is shown to depend critically on the initial function, and the parameters alpha > 1 and lambda > 0. This dependence is in full agreement with earlier numerical results by Barenblatt and Prostokishin [1].