The authors prove that the nonlinear parabolic partial differential equation
∂u∂t=∑i,j=1n∂2∂xi∂xjφij(u)−f(u) with homogeneous Dirichlet boundary conditions and a nonnegative initial condition has a nonnegative generalized solution u. They also give necessary and sufficient conditions on the constitutive
functions φij and f which ensure the existence of a time t0>0 for which u vanishes for all t≥t0.