Based on a self-similar analysis of the Fujita type global solvability, we obtain estimates for a solution and fronts (a free boundary) of the Cauchy problem for a doubly nonlinear nondivergence-form parabolic equation with variable density. We consider the case where the absorpiton depends on time and study the asymptotic behavior of the finite self-similar solution depending on numerical parameters characterizing a nonlinear medium. The main results are illustrated by numerical examples.
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References
M. Aripov and S. Sadullaeva, Computer Simulation of Nonlinear Diffusion Processes [in Russian], University Press, Tashkent (2020).
A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-Up in Quasilinear Parabolic Equations, Walter de Grueter, Berlin (1995).
M. Aripov, A. S. Matyakubov, J. O. Khasanov, and M. M. Bobokandov, “Mathematical modeling of double nonlinear problem of reaction diffusion in non-divergence form with a source and variable density,” J. Phys., Conf. Ser. 2131, No. 3, Article ID 032043 (2021).
J. Shao et al., “A new non-divergence diffusion equation with variable exponent for multiplicative noise removal,” Nonlinear Anal., Real World Appl. 56, Article ID 103166 (2020).
C. P. Wang and J. X. Yin, “Shrinking self-similar solutions of a nonlinear diffusion equation with nondivergence form,” J. Math. Anal. Appl. 289, No. 2, 387–404 (2004).
D. Andreucci and A. F. Tedeev, “Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity,” Adv. Differ. Equ. 5, No. 7–9, 833–860 (2000).
Z. Xiang, C. Mu, and X. Hu, “Support properties of solutions to a degenerate equation with absorption and variable density,” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 7, 1940–1953 (2008).
V. A. Galaktionov and J. R. King, “On the behavior of blow-up interfaces for an inhomogeneous filtration equation,” IMA J. Appl. Math. 57, No. 1, 53–77 (1996). https://doi.org/10.1093/imamat/57.1.53
M. Aripov, A. S. Matyakubov, and M. M. Bobokandov, “Cauchy problem for the heat dissipation equation in non-homogeneous medium,” AIP Conf. Proc. 2781, No. 1, Article ID 020027 (2023). https://doi.org/10.1063/5.0144807
S. Zhou, X. Tang, and C. Yang, “A doubly degenerate diffusion equation not in divergence form with gradient term,” Bound. Value Probl. 2016, Article ID 126 (2016).
C. Jin and J. Yin, “Asymptotic behavior of solutions for a doubly degenerate parabolic non-divergence form equation,” Rocky Mt. J. Math. 47, No. 2, 479–510 (2017).
H. Fujita, “On the blowing up of solutions of the Cauchy problem for ut = Δu + u1+σ,” J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109–124 (1966).
V. A. Galaktionov and H. A. Levine, “A general approach to critical Fujita exponents and systems,” Nonlinear Anal., Theory Methods Appl. 34, No. 7, 1005–1027 (1998).
A. V. Martynenko, A. F. Tedeev, and V. N. Shramenko, “The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data,” Izv. Math. 76, No. 3, 563–580 (2012).
A. V. Martynenko and A. F. Tedeev, “Cauchy problem for a quasilinear parabolic equation with a source and an inhomogeneous density,” Comput. Math. Math. Phys. 47, No. 2, 238–248 (2007).
D. Andreucci and A. F. Tedeev, “A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary,” J. Math. Anal. Appl. 231, No. 2, 543–567 (1999).
V. A. Galaktionov and J. L. Vazquez, “Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach,” J. Funct. Anal. 100, No. 2, 435–462 (1991).
M. Aripov, A. Mukimov, and B. Mirzayev, “To asymptotic of the solution of the heat conduction problem with double nonlinearity with absorption at a critical parameter,” Int. J. Innov. Technol. Eng. DOI: 1035940/ijiteeA4455.119119 (2019).
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Aripov, M., Bobokandov, M. The Cauchy Problem for a Doubly Nonlinear Parabolic Equation with Variable Density and Nonlinear Time-Dependent Absorption. J Math Sci 277, 355–365 (2023). https://doi.org/10.1007/s10958-023-06840-0
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DOI: https://doi.org/10.1007/s10958-023-06840-0