What is it about?
The paper shows how variable density and time-dependent forcing shift the classical Fujita-type picture, and it provides constructive comparison profiles that are useful beyond theory: they guide nondimensionalization and yield diagnostic power laws.
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Why is it important?
The study is significant because it develops a coherent analytical picture for a time-dependent, weighted, doubly nonlinear parabolic Cauchy problem with source/absorption. The interplay between (i) variable density, (ii) nonlinear mobility, and (iii) the time-modulated reaction term produces regimes where solutions are weak, localized, and potentially nonunique—features that mirror realistic transport in heterogeneous media. By combining weak-solution notions with comparison principles, explicit self-similar reductions, and profile asymptotics, the paper identifies critical parameter restrictions (of Fujita type) governing global versus non-global behaviour. It characterises the geometry of propagation via compact-support/free-boundary-type behaviour.
Perspectives
Practically, the paper’s similarity profiles and bounds look like a ready-made toolkit for parameter identification and model calibration: measuring front position, decay rates, or growth rates can be matched to predicted scaling laws to estimate effective exponents and density weights in experiments or simulations. On the computational side, the emphasis on choosing an initial approximation consistent with nonlinear localisation is exactly what robust solvers need; it would be interesting to formalise this into an “admissible initialisation” principle for iterative schemes in degenerate media. A further step is to extend the approach to coupled systems (cross-diffusion, multi-species filtration) and to boundary-driven settings, where the same nonlinearity–degeneracy mechanisms appear but with additional interface dynamics.
Mr. Makhmud Bobokandov
National University of Uzbekistan
Read the Original
This page is a summary of: Cauchy problem for the double nonlinear parabolic equation not in divergent form with a time-dependent source or absorption, Carpathian Mathematical Publications, December 2025, Vasyl Stefanyk Precarpathian National University,
DOI: 10.15330/cmp.17.2.693-705.
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