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Existence of solutions to a diffusive shallow medium equation.

Verena Bögelein1, Nicolas Dietrich1, Matias Vestberg2

  • 1Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria.

Journal of Evolution Equations
|November 15, 2021
PubMed
Summary

This study proves the existence of weak solutions for the shallow medium equation using a novel approximation method. The findings establish a foundation for further analysis of this complex mathematical model.

Keywords:
Doubly nonlinear parabolic equationsExistenceShallow medium equation

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Area of Science:

  • Mathematical analysis
  • Partial differential equations

Background:

  • The shallow medium equation is a complex mathematical model.
  • Understanding its solutions is crucial for various scientific applications.

Purpose of the Study:

  • To establish the existence of weak solutions for the shallow medium equation.
  • To develop a rigorous mathematical framework for analyzing this equation.

Main Methods:

  • The study employs an approximation argument.
  • Coefficients of the equation are truncated from above and below.
  • Convergence of solutions for the truncated problem is proven.

Main Results:

  • The existence of weak solutions to the shallow medium equation is established.
  • The approximation method successfully converges to a solution of the original equation.

Conclusions:

  • The developed approximation technique provides a valid method for proving the existence of weak solutions.
  • This work contributes to the theoretical understanding of the shallow medium equation.