Two-parameter asymptotics in a bisingular Cauchy problem for a parabolic equation

The Cauchy problem for a quasilinear parabolic equation with a small parameter $\varepsilon$ at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a “stretched” variable $x/\rho$, where $\rho$ is another small parameter. This problem statement is of interest in applications as a model of propagation of nonlinear waves in physical systems in the presence of small dissipation. In the case corresponding to a compression wave, asymptotic solutions of the problem are constructed in the parameters $\varepsilon$ and $\rho$ independently tending to zero. It is assumed that $\varepsilon/\rho\to 0$. Far from the line of discontinuity of the limit solution, asymptotic solutions are constructed in the form of series in powers of $\varepsilon$ and $\rho$. In a small domain of linear approximation, an asymptotic solution is constructed in the form of a series in powers of the ratio $\rho/\varepsilon$. The coefficients of the inner expansion are found from a recurrence chain of initial value problems. The asymptotics of these coefficients at infinity is studied. The time of reconstruction of the scale of the inner space variable is found.