Harnack's inequality for doubly nonlinear equations of slow diffusion type

Area of Science:

• Partial Differential Equations
• Nonlinear Analysis
• Mathematical Physics

Background:

• Doubly nonlinear parabolic equations are crucial in modeling various physical phenomena.
• Understanding the behavior of non-negative weak solutions is essential for theoretical and applied mathematics.
• Previous research has focused on specific ranges of exponents, leaving gaps in the analysis.

Purpose of the Study:

• To establish a Harnack inequality for non-negative weak solutions to a specific class of doubly nonlinear parabolic equations.
• To extend the analysis to the full range of the slow diffusion case, encompassing all relevant exponents.
• To provide a rigorous mathematical framework for analyzing these equations under given p-ellipticity and growth conditions.

Main Methods:

• Utilizing techniques from the theory of partial differential equations.
• Applying methods related to elliptic and parabolic regularity.
• Developing and adapting inequalities for weak solutions in nonlinear settings.

Main Results:

• A novel Harnack inequality is proven for non-negative weak solutions.
• The inequality holds for doubly nonlinear parabolic equations with a p-elliptic vector field.
• The results cover the entire slow diffusion regime, including all exponents p and q where p > 1 and q > 1.

Conclusions:

• The established Harnack inequality provides a significant advancement in the study of doubly nonlinear parabolic equations.
• This work contributes to a deeper understanding of the regularity properties of solutions in the slow diffusion case.
• The findings have potential implications for mathematical modeling in physics and engineering where such equations arise.