Some free boundary problems involving non-local diffusion and aggregation

Area of Science:

• Mathematical Physics
• Partial Differential Equations
• Diffusion Theory

Background:

• Degenerate parabolic equations are crucial for modeling diffusion processes.
• Recent advancements focus on anomalous diffusion and aggregation phenomena.
• Free boundaries present significant challenges in these models.

Purpose of the Study:

• To present recent progress in the study of evolution processes governed by degenerate parabolic equations.
• To investigate models incorporating anomalous diffusion with long-range effects.
• To analyze the interplay between diffusion and aggregation leading to complex equilibria.

Main Methods:

• Utilizing fractional operators and kernels with large tails to model anomalous diffusion.
• Developing mathematical frameworks for degenerate parabolic equations with free boundaries.
• Analyzing long-term behaviors and equilibria in combined diffusion-aggregation systems.

Main Results:

• Demonstrated progress in understanding evolution processes with free boundaries.
• Characterized anomalous diffusion through fractional operators and large-tail kernels.
• Identified delicate long-term equilibria arising from diffusion-aggregation effects.

Conclusions:

• The study advances the understanding of complex diffusion phenomena.
• New insights into modeling anomalous diffusion and aggregation are provided.
• Further research is needed to fully describe the emergent long-term equilibria.