Asymptotic behavior of generalized self-similar solutions for a nonlinear hybrid problem of porous medium equations
Keywords:
porous medium equation, generalized self-similar solution, blow-up, global existence, uniquenessAbstract
The present paper investigates the asymptotic behavior of positive generalized self-similar solutions for a nonlinear hybrid problem involving nth-order derivative porous medium equations. We provide sufficient conditions for the existence and uniqueness of weak solutions that have compact support and dynamic characteristics. Furthermore, we establish the behavior of these solutions by examining a specific set of variables and their signs, which must meet certain conditions to determine whether the solutions exist globally or locally in time.
References
Barenblatt, G.I. "On some unsteady motions of a liquid or a gas in a porous medium." Prikl. Mat. Mekh. 16 (1952): 67-78.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Basti, B., and N. Benhamidouche. "Global existence and blow-up of generalized self-similar solutions to nonlinear degenerate diffusion equation not in divergence form." Appl. Math. E-Notes 20 (2020): 367-387.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Benhamidouche, N. "Exact solutions to some nonlinear PDEs, traveling profiles method." Electron. J. Qual. Theory Differ. Equ. 15 (2008): 1-7.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Benhamidouche, N., and Y. Arioua. "New method for constructing exact solutions to non-linear PDEs." Int. J. Nonlinear Sci. 7 (2009): 395-398.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Caffarelli, L., and J.L. Vàzquez. "Viscosity solutions for the porous medium equation." Proc. Sympos. Pure Math. 65 (1999): 13-26.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Gilding, B.E., and L.A. Peletier. "On a class of similarity solution of the porous medium equation I." J. Math. Anal. Appl. 55 (1976): 351-364.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Granas, A., and J. Dugundji. Fixed Point Theory. New York: Springer-Verlag, 2003.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Guo, D., and V. Lakshmikantham. Nonlinear Problems in Abstract Cones. New York: Academic Press, 1988.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Hulshof, J., and J.L. Vàzquez. "Self-similar solutions of the second kind for the modified porous medium equation." Eur. J. Appl. Math. 5 (1994): 391-403.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Hulshof, J., and J.L. Vàzquez. "Maximal viscosity solutions of the modified porous medium equation and their asymptotic behavior." Eur. J. Appl. Math. 7 (1996): 453-471.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Ibragimov, N.H. CRC Handbook of Lie Group Analysis of Differential Equations. Vol.1: Symmetries, Exact Solutions and Conservation Laws. Boca Raton: CRC Press, 1994.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Olver, P.J. Applications of Lie Groups to Differential Equations. New York: Springer-Verlag, 1986.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Polyanin, A.D., and V.F. Zaitsev. Handbook of Nonlinear Partial Differential Equations. Boca Raton: Chapman & Hall/CRC, 2004.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Vàzquez, J.L. The Porous Medium Equation: Mathematical Theory. Oxford: Oxford University Press, 2007.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Wang, C., and Jingxue. "Shrinking self-similar solution of a nonlinear diffusion equation with nondivergence form." J. Math. Anal. Appl. 289 (2004): 387-404.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Wydawnictwo Naukowe Uniwersytetu Komisji Edukacji Narodowej
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
