What is it about?

Non‑divergence, nonlinear parabolic equation with a source. Derive Fujita‑type conditions that distinguish between global solvability and finite‑time blow‑up. Construct and analyse self‑similar solutions that capture the leading large‑time behavior.

Featured Image

Why is it important?

Combining two nonlinearities (in u and gradient u) in non‑divergence form breaks the standard energy methods. Establishing precise critical exponents and long‑time patterns fills a gap in the theory of degenerate parabolic PDEs and extends Fujita’s classical blow‑up vs. global existence dichotomy. Numerical and practical relevance. Understanding self‑similar attractors guides the design of robust numerical schemes and helps engineers predict whether a process stabilises or blows up under nonlinear feedback and density variations

Perspectives

Refined self‑similar analysis: Explore the existence/uniqueness of self‑similar profiles in borderline (“critical”) cases and derive their stability rates. Advanced numerics: Develop adaptive finite‑difference or finite‑volume methods respecting the non‑divergence structure and verify convergence to the predicted asymptotic profiles. Coupled systems: Extend to reaction‑diffusion systems (e.g., coupling u with temperature or chemical concentration) where multiple interacting fields share similar degeneracies. Irregular densities & domains: Study how non‑radial or anisotropic density variations (including obstacles or layers) affect blow‑up thresholds and long‑time decay. Singular sources: Investigate time‑singular or spatially concentrated sources (e.g., Dirac‑type forcing) to model localized injections or sinks in nonlinear media.

Mr. Makhmud Bobokandov
National University of Uzbekistan

Read the Original

This page is a summary of: Asymptotic behavior of solutions for a doubly nonlinear parabolic non-divergence form equation with density, January 2024, American Institute of Physics,
DOI: 10.1063/5.0199587.
You can read the full text:

Read

Resources

Contributors

The following have contributed to this page