The Cauchy problem for a quasilinear parabolic equation with a large initial gradient and low viscosity

The Cauchy problem for a quasilinear parabolic equation with a small parameter $\varepsilon$ multiplying the highest derivative is considered. The derivative of the initial function is on the order of $O(1/\rho)$, where $\rho$ is another small parameter. Asymptotic expansions of the solution in powers of $\varepsilon$ and $\rho$ are constructed in various forms.