M. Bertsch et P. Bisegna, BLOW-UP OF SOLUTIONS OF A NONLINEAR PARABOLIC EQUATION IN DAMAGE MECHANICS, European journal of applied mathematics, 8, 1997, pp. 89-123
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 configura
tion, the blow-up set, corresponding to the portion of the material wh
ich breaks at the blow-up time, is an interval of nonzero measure. In
a special but physically relevant case the problem reduces to the stud
y 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, an
d the size of the blow-up set is shown to depend critically on the ini
tial function, and the parameters alpha > 1 and lambda > 0. This depen
dence is in full agreement with earlier numerical results by Barenblat
t and Prostokishin [1].