  We study the asymptotic speed of traveling fronts of the scalar reaction
diffusion for positive reaction terms and with a diffusion coefficient
depending nonlinearly on the concentration and on its gradient. We restrict our
study to diffusion coefficients of the form $D(u,u_x) = m u^{m-1} u_x^{m(p-2)}$
for which existence and convergence to traveling fronts has been established.
We formulate a variational principle for the asymptotic speed of the fronts.
Upper and lower bounds for the speed valid for any $m\ge0, p\ge 1$ are
constructed. When $m=1, p=2$ the problem reduces to the constant diffusion
problem and the bounds correspond to the classic Zeldovich Frank-Kamenetskii
lower bound and the Aronson-Weinberger upper bound respectively. In the special
case $m(p-1) = 1$ a local lower bound can be constructed which coincides with
the aforementioned upper bound. The speed in this case is completely determined
in agreement with recent results.
