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A comparison principle for doubly nonlinear parabolic partial differential equations.

Verena Bögelein1, Michael Strunk1

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

Annali Di Matematica Pura Ed Applicata
|March 12, 2024
PubMed
Summary

This study establishes a comparison principle for doubly nonlinear parabolic partial differential equations, proving solution uniqueness and demonstrating that weak solutions are also viscosity solutions, even with only positive lateral boundary data.

Keywords:
Comparison principleDoubly nonlinear parabolic PDE

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Area of Science:

  • Partial Differential Equations
  • Nonlinear Analysis
  • Mathematical Physics

Background:

  • Doubly nonlinear parabolic partial differential equations model complex physical phenomena.
  • Existing methods often require lower bounds on solutions across the entire domain.
  • The prototype equation considered is .

Purpose of the Study:

  • To derive a novel comparison principle for non-negative weak sub- and super-solutions.
  • To relax the condition of a lower bound on solutions to only require strictly positive lateral boundary data.
  • To explore applications of this principle, including solution uniqueness and classification.

Main Methods:

  • Derivation of a comparison principle for weak solutions.
  • Analysis of doubly nonlinear parabolic partial differential equations.
  • Application of the principle to Cauchy-Dirichlet problems.

Main Results:

  • A new comparison principle is established for non-negative weak sub- and super-solutions.
  • Uniqueness of non-negative weak solutions to the Cauchy-Dirichlet problem is proven.
  • It is demonstrated that any weak solution is also a viscosity solution.

Conclusions:

  • The derived comparison principle offers a more flexible tool for analyzing these equations.
  • The results contribute to a deeper understanding of the behavior of solutions.
  • The findings have implications for the study of related nonlinear partial differential equations.