On finite diffusing speed for uniformly degenerate quasilinear parabolic equations (Q1083612)

The author studies a quasilinear parabolic equation in divergence form, which is uniformly degenerate - which means that the quadratic form associated with the diffusion coefficient satisfies \[ \nu (| u|) | \xi |^ 2\leq a_{ij}(x,t,u) \xi_ i \xi_ j\leq \Lambda \nu (| u|) \xi |^ 2, \] where \(\Lambda\) is a positive constant and \(\nu\) is a continuous function, which is positive for positive arguments and vanishes at zero. In addition, \(\nu\) is required to satisfy an additional assumption which could be referred to as a slow diffusion requirement. The object of the paper is to establish a pointwise estimate for the solution implying the finite propagation speed property. The proof makes use of the maximum principle and of differential inequalities for appropriate functionals of the solution.