Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption

TL;DR

This paper derives short-time interface asymptotics for a nonlinear degenerate reaction-diffusion equation modeling turbulent filtration with absorption, supported by numerical validation using advanced WENO schemes.

Contribution

It provides the first short-time asymptotic formulas for interfaces in this class of equations and offers a comprehensive classification based on nonlinearity parameters.

Findings

Derived explicit interface asymptotics for the equation.

Classified solution behaviors depending on parameters.

Validated analytical results with numerical simulations.

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 strong absorption \[ u_t=\Big(|(u^{m})_x|^{p-1}(u^{m})_x\Big)_x-bu^{\beta}, \, mp>1, \, \beta >0. \] Full classification is pursued in terms of the nonlinearity parameters $m, p,\beta$ and asymptotics of the initial function near its support. Numerical analysis using a weighted essentially nonoscillatory (WENO) scheme with interface capturing is implemented, and comparison of numerical and analytical results is presented.