Global existence results for coupled nonlinear parabolic equations with weighted coefficients
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In this paper, we investigate a class of nonlinear weighted parabolic systems describing the coupled
dynamics of two interacting scalar fields. We establish sufficient conditions for the global existence of
weak solutions in appropriate weighted Sobolev spaces by employing energy estimates and integral
inequalities. Furthermore, we develop a numerical scheme based on the Peaceman-Rachford splitting
method combined with the Thomas algorithm to approximate the solutions efficiently. The proposed
computational framework is implemented and illustrated with two and three-dimensional numerical
simulations, including dynamic surface plots and animated profiles. The results demonstrate the
qualitative features of the global solution, confirm the analytical findings, and provide additional
insight into the interplay between nonlinear diffusion, weighted heterogeneity, and inter-component
coupling.
1. Abdugapor Kh., Mamatov A. Modeling of double nonlinear thermal conductivity processes in two-
dimensional domains using solutions of an approximately self-similar. AIP Conference Proceedings, 2023,
vol. 2781, p. 020067.
2. Ansgar J. Cross-Diffusion systems with entropy structure. In Proceedings of Equadiff 14. Conference on
Differential Equations and Their Applications, Bratislava, 2017, pp. 24–28.
3. Aripov M., Bobokandov M., Uralov N. Analysis of Double Nonlinear Parabolic Crosswise-Diffusion
Systems with Time-Dependent Nonlinearity Absorption. In International Conference on Thermal
Engineering, 2024, vol. 1, no. 1.
4. Carrillo J.A., Di Francesco J.A., Figalli M., Laurent A., Slepˇ cev D. Global-in-time weak measure solutions
and finite-time aggregation for nonlocal interaction equations. Duke Mathematical Journal, 2011, pp. 229–
271.
5. Deng L., Shang X. Doubly degenerate parabolic equation with time-dependent gradient source and initial
data measures. Journal of Function Spaces, 2020, vol. 11, no. 2, pp. 25–32.
6. DiBenedetto E. On the local behavior of solutions of degenerate parabolic equations with measurable
coefficients. Annali della Scuola Normale Superiore di Pisa, 1986, vol. 13, no. 3, pp. 487–535.
7. Gander J.M., Stuart M.A. Analysis of the Peaceman-Rachford splitting scheme applied to a class of
nonlinear evolution equations. SIAM Journal on Numerical Analysis, 2008, vol. 46, no. 4, pp. 2097–2113.
8. Giachetti D., Porzio M. Global existence for nonlinear parabolic equations with a damping term.
Communications on Pure and Applied Analysis, 2019, vol. 8, no. 3, pp. 923–653.
9. Hui K. Another proof for the removable singularities of the heat equation. Proc. Amer. Math. Soc., 2010,
vol. 138, no. 7, pp. 2397–2402.
10. Ketcheson I.D. Relaxation Runge-Kutta methods: Fully implicit schemes for hyperbolic problems. SIAM
Journal on Scientific Computing, 2019, vol. 41, no. 2, pp. A1063–A1085.
11. Kok F. Global and regional importance of the mineral dust cycle: A review. Atmospheric Environment,
2017, vol. 161, pp. 38–57.
12. Ladyzhenskaya O.A., Ural’tseva N.N. Linear and Quasilinear Equations of Parabolic Type. Academic
Press, New York-London, 1968.
13. LeVeque J. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 2007, vol.
47, no. 6.
14. Lions J. Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires. Dunod, Paris, 1969.
15. Martynenko A., Tedeev A., Shramenko V. The Cauchy problem for a degenerate parabolic equation with
inhomogeneous density and source in the class of slowly decaying initial data. Izv. Math., 2012, vol. 76,
no. 3, pp. 563–580.
16. Mamatov A. Properties of solutions of a nonlinear reaction-diffusion system with variable density and
source. Uzbek Mathematical Journal, 2020, vol. 66, no. 4, pp. 70–79.
17. Matyakubov A., Raupov D. Explicit estimate for blow-up solutions of nonlinear parabolic systems of
non-divergence form with variable density. AIP Conference Proceedings, 2023, vol. 2781, p. 020055.
18. Nicolosi F., Skrypnik I.I., Skrypnik I.V. Removable isolated singularities for solutions of quasilinear
parabolic equations. Ukr. Math. Bull., 2009, vol. 6, pp. 208–234.
19. Nurumova A. Blow-up case for some nonlinear differential inequalities. Journal of Advanced Research in
Dynamical and Control Systems, 2020, vol. 12, no. 3, pp. 147–158.
20. Punzo F. Well-posedness of Degenerate Elliptic and Parabolic Problems. Ph.D. thesis, 2008.
21. Reyes G., Vazquez J. The Cauchy problem for the inhomogeneous porous medium equation. Netw. Heterog.
Media, 2006, vol. 1, pp. 337–351.
22. Seinfeld J.H., Pandis S.H. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change.
Wiley-Interscience, 2012, pp. 430–520.
23. Shao J., Guo Z., Shan X., Zhang Ch., Wu B. A new non-divergence diffusion equation with variable
exponent for multiplicative noise removal. Nonlinear Analysis: Real World Applications, 2020, vol. 56, p.
103166.
24. Smith A., Zhao L. Numerical methods for nonlinear degenerate parabolic PDEs with applications to
diffusion processes. Journal of Computational Physics, 2023, p. 111797.
25. VГЎzquez J.L. The Porous Medium Equation: Mathematical Theory. Oxford University Press, Oxford,
2007, pp. 145–165.
26. VГЎzquez J.L. Smoothing and decay estimates for nonlinear diffusion equations: equations of porous
medium type. Oxford University Press, Oxford, 2006.
27. Zhang H. The self-similar solutions of a diffusion equation. Journal of WSEAS Transactions on
Mathematics, 2011, vol. 12, no. 3, pp. 345–356.
28. Zhang H., Xiamen P. The nonexistence of the solution for quasilinear parabolic equation related to the
p-laplacian equation. Journal of WSEAS Transactions on Mathematics, 2012, vol. 11, no. 4, pp. 679–688.
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