# Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption. II. Fast Diffusion Case

## TL;DR

This paper analyzes the short-time behavior of interfaces and solutions for a nonlinear reaction-diffusion equation modeling turbulent filtration with fast diffusion and absorption, providing classifications based on parameters and initial conditions.

## Contribution

It extends previous work by classifying the short-time asymptotics of interfaces and solutions for the fast diffusion case of the equation.

## Key findings

- Derived asymptotic formulas for interface behavior
- Classified solutions based on parameters and initial data
- Extended understanding of interface evolution in turbulent filtration models

## Abstract

We prove the short-time asymptotic formula for the interfaces and local solutions near the interfaces for the nonlinear double degenerate reaction-diffusion equation of turbulent filtration with fast diffusion and strong absorption \[ u_t=(|(u^{m})_x|^{p-1}(u^{m})_x)_x-bu^{\beta}, \, 0<mp<1, \, \beta >0. \] Full classification is pursued in terms of the nonlinearity parameters $m, p,\beta$ and asymptotics of the initial function near its support. In the case of an infinite speed of propagation of the interface, the asymptotic behavior of the local solution is classified at infinity. A full classification of the short-time behavior of the interface function and the local solution near the interface for the slow diffusion case ($mp>1$) was presented in $\textit{Abdulla et al., Math. Comput. Simul., 153(2018), 59-82}$.