Propagation versus constancy of support in the degenerate parabolic equation $u_t=f(u)\Delta u$

Abstract

A weak solution concept for the Dirichlet problem in bounded domains for the degenerate parabolic equation $u_t = f(u)\Delta u$ is presented. It is shown that if $\int_0^1 \frac{ds}{f(s)}<\infty$ then each nontrivial nonnegative weak solution eventually becomes positive, while if $\int_0^1 \frac{ds}{f(s)} = \infty$ then all weak solutions have their support constant in time.