This paper presents a thorough analytical and numerical investigation of a strongly nonlinear, coupled reaction-diffusion system formulated for modeling atmospheric dust transport in heterogeneous media. The governing equations represent a doubly nonlinear parabolic system of the Fisher-KPP type, containing generalized p-Laplacian diffusion operators, nonlinear interphase exchange kinetics, and coefficients that are dependent on space and time. To analyze the qualitative behavior of the system, we use a self-similar reduction, a comparison of the corresponding behavior with compact-support Barenblatt-type profiles. Using variational methods and comparison principles, it is shown that weak solutions exist and are unique in the right Sobolev spaces. Moreover, to properly represent the intricate spatiotemporal dynamics and the limited propagation speed property of the degenerate diffusion terms, we employ an effective numerical method. Our numerical simulations establish the theoretical similarity solutions and demonstrate the model's ability to represent localized nonlinear diffusion effects, complex phase interactions, and actual dust concentration patterns, thereby offering an effective mathematical and predictive structure for environmental transport mechanisms.
A. Berezovski, J. Engelbrecht, and G. A. Maugin, Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. Singapore: World Scientific, 2008.
A. Smith and L. Zhao, “Numerical methods for nonlinear degenerate parabolic PDEs with applications to diffusion processes,” Journal of Computational Physics, vol. 493, p. 111797, 2023.
Sh. Kh. Sultonov, T. R. Nurmukhamedov, D. Kh. Sultanova, O. A. Turdiev, and A. A. Azimov, “Development of the meteorological block algorithm of the electric heating system for turnouts,” in S. M. Sultanova, N. U. Babakhanova, S. S. Shaumarov, and M. N. Masharipov, Eds., Sustainable Development of Transport: Economy, Transformation, Logistics and ESG Agenda, Lecture Notes in Networks and Systems, vol. 1547. Cham, Switzerland: Springer, 2025, [Online]. Available: https://doi.org/10.1007/978-3-031-99028-1_3.
Z. Zhang and Y. Li, “Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source,” Applied Mathematics and Optimization, vol. 93, no. 2, Art. no. 28, 2026.
N. Kato, M. Misawa, K. Nakamura, and Y. Yamaura, “Existence for doubly nonlinear fractional p-Laplacian equations,” Annali di Matematica Pura ed Applicata, vol. 203, no. 6, pp. 2481-2527, 2024.
X. Pan, “Superlinear degradation in a doubly degenerate nutrient taxis system,” Nonlinear Analysis: Real World Applications, vol. 77, Art. no. 104040, 2024.
G. Li and M. Winkler, “Continuous solutions for a two-dimensional cross-diffusion problem involving doubly degenerate diffusion and logistic proliferation,” Analysis and Applications, vol. 23, no. 4, pp. 489-510, 2025.
O. A. Turdiev, M. Masharipov, M. M. Khalil, and M. S. Mohammed, “Comparison of CRC16 and PNC16 models to identify errors in Python,” in Proceedings of the International Conference on Applied Innovation in IT, 2025, [Online]. Available: https://doi.org/10.25673/120441.
Y. Xiang, “Boundedness and large-time behavior in a two-species doubly degenerate diffusion chemotaxis system with logistic proliferation,” Nonlinear Analysis: Real World Applications, vol. 89, Art. no. 104525, 2026.
J. Zheng and Y. Ke, “Global existence and eventual smoothness for a 2-dimensional Keller-Segel system with nonlinear logistic source,” Journal of Differential Equations, vol. 437, Art. no. 113293, 2025.
C. Xu, “Global boundedness in a quasilinear chemotaxis-consumption system with degenerate signal-dependent motility and logistic source,” Zeitschrift für Angewandte Mathematik und Physik, vol. 75, no. 6, 2024.
M. Winkler, “L∞ bounds in a two-dimensional doubly degenerate nutrient taxis system with general cross-diffusive flux,” Journal of Differential Equations, vol. 400, pp. 423-456, 2024.
L. Wang and C. Mu, “Global solvability and eventual smoothness in a degenerate migration-consumption system with logistic source,” Calculus of Variations and Partial Differential Equations, vol. 64, no. 8, 2025.
N. Chen, P. Wang, and F. Li, “Global existence and blow-up phenomena for the doubly nonlinear diffusion equation with nonlinear Neumann boundary conditions,” Journal of Applied Analysis and Computation, vol. 14, no. 3, pp. 1467-1484, 2024.
Y. Tao and M. Winkler, “Suppression of blow-up by local anisotropy of signal production in the Keller-Segel system,” Journal des Mathématiques Pures et Appliquées, vol. 205, Art. no. 103795, 2026.
A. Mukhamadiyev, J. Urunbaev, M. Bobokandov, Z. Rakhmonov, and T. Khujakulov, “A self-similar analysis of the solutions to the cross-diffusion system,” Mathematics, vol. 14, no. 1, Art. no. 83, 2026.
M. Misawa and Y. Yamaura, “Expansion of positivity for the doubly nonlinear parabolic fractional type equations,” Mathematische Annalen, vol. 393, no. 2, pp. 2561-2629, 2025.
K. Baghaei, “Blow-up in finite time for a pseudo-parabolic equation with variable exponents,” Applicable Analysis, vol. 104, no. 12, pp. 2367-2382, 2025.
M. S. Mohammed, A. M. S. Ahmed, A. A. A. Al Wahab, S. J. Mohammed, A. T. I. Salman, and O. A. Turdiev, “Advanced electronic circuit design for visual classification modeling using CNN,” Journal of Internet Services and Information Security, vol. 15, no. 2, pp. 774-790, May 2025, [Online]. Available: https://doi.org/10.58346/JISIS.2025.I2.051.
W. Lyu and J. Hu, “The convergence rate of solutions in chemotaxis models with density-suppressed motility and logistic source,” Nonlinear Differential Equations and Applications, vol. 31, no. 5, 2024.
M. Aripov, M. Bobokandov, and M. Mamatkulova, “Analysis of a double nonlinear diffusion equation in inhomogeneous medium,” Journal of Mathematical Sciences, vol. 289, no. 5, pp. 657-669, 2025.
Z. Yu and Y. Li, “Global solvability in a doubly degenerate migration system with nonlinear consumption,” Discrete and Continuous Dynamical Systems - Series B, vol. 30, no. 8, pp. 3042-3054, 2025.
Z. Zhang and Y. Li, “Boundedness in a two-dimensional doubly degenerate nutrient taxis system,” Mathematical Models and Methods in Applied Sciences, vol. 36, no. 3, pp. 527-559, 2026.
D. Wu, “Refined existence theorems for doubly degenerate chemotaxis-consumption systems with large initial data,” Nonlinear Differential Equations and Applications, vol. 31, no. 6, 2024.
A. Gouasmia, “Uniqueness results for mixed local and nonlocal equations with singular nonlinearities and source terms,” Mathematische Nachrichten, 2026.
M. Vestberg, “Existence, comparison principle and uniqueness for doubly nonlinear anisotropic evolution equations,” Journal of Evolution Equations, vol. 25, no. 1, Art. no. 14, 2025.
I. Dahi and A. Sidi, “Renormalized solutions for a triply nonlinear nonlocal thermistor problem with L¹ data,” Annali dell’Università di Ferrara, vol. 72, no. 2, Art. no. 21, 2026.
W. Lyu and J. Jiang, “Global boundedness and stabilization of solutions for a chemotaxis system with acceleration and logistic source,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2025.
P. Rani and J. Tyagi, “A quasilinear chemotaxis-haptotaxis system: Existence and blow-up results,” Journal of Differential Equations, vol. 402, pp. 180-217, 2024, [Online]. Available: https://doi.org/10.1016/j.jde.2024.04.034.
S. Uchida, “A doubly nonlinear parabolic equation with nonlinear perturbation under relaxed growth and exponent conditions,” Nonlinear Analysis: Real World Applications, vol. 90, Art. no. 104549, 2026.
