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 ɛ multiplying the highest derivative is considered. The derivative of the initial function is on the order of O(1/ρ), where ρ is another small parameter. Asymptotic expansions of the solution in powers of ɛ and ρ are constructed in various forms.