Abstract
This paper is concerned with group invariant solutions for fast diffusion equations in symmetric domains. First, it is proved that the group invariance of weak solutions is inherited from initial data. After briefly reviewing previous results on asymptotic profiles of vanishing solutions and their stability, the notions of stability and instability of group invariant profiles are introduced under a similarly invariant class of perturbations, and moreover, some stability criteria are exhibited and applied to symmetric domain (e.g., annulus) cases.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aftalion, A., Pacella, F.: Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. C. R. Math. Acad. Sci. Paris 339, 339–344 (2004)
Akagi, G.: Energy solutions of the Cauchy-Neumann problem for porous medium equations. Discrete Contin. Dyn. Syst. suppl., 1–10 (2009)
Akagi, G., Kajikiya, R.: Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations. Manuscr. Math. doi:10.1007/s00229-012-0583-9
Akagi, G., Kobayashi, J., Ôtani, M.: Principle of symmetric criticality and evolution equations. Discrete Contin. Dyn. Syst. suppl., 1–10 (2003)
Bénilan, P., Crandall, M.G.: The continuous dependence on φ of solutions of u t −Δφ(u)=0. Indiana Univ. Math. J. 30, 161–177 (1981)
Berryman, J.G., Holland, C.J.: Nonlinear diffusion problem arising in plasma physics. Phys. Rev. Lett. 40, 1720–1722 (1978)
Berryman, J.G., Holland, C.J.: Stability of the separable solution for fast diffusion. Arch. Ration. Mech. Anal. 74, 379–388 (1980)
Bonforte, M., Grillo, G., Vazquez, J.L.: Behaviour near extinction for the fast diffusion equation on bounded domains. J. Math. Pures Appl. 97, 1–38 (2012)
Brézis, H.: Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis, pp. 101–156. Academic Press, New York (1971)
Byeon, J.: Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli. J. Differ. Equ. 136, 136–165 (1997)
Coffman, C.V.: A nonlinear boundary value problem with many positive solutions. J. Differ. Equ. 54, 429–437 (1984)
DiBenedetto, E., Kwong, Y., Vespri, V.: Local space-analyticity of solutions of certain singular parabolic equations. Indiana Univ. Math. J. 40, 741–765 (1991)
Feiresl, E., Simondon, F.: Convergence for semilinear degenerate parabolic equations in several space dimension. Dyn. Partial Differ. Equ. 12, 647–673 (2000)
Gazzola, F., Weth, T.: Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ. Integral Equ. 18, 961–990 (2005)
Kwong, Y.C.: Asymptotic behavior of a plasma type equation with finite extinction. Arch. Ration. Mech. Anal. 104, 277–294 (1988)
Li, Y.Y.: Existence of many positive solutions of semilinear elliptic equations in annulus. J. Differ. Equ. 83, 348–367 (1990)
Ni, W.-M.: Uniqueness of solutions of nonlinear Dirichlet problems. J. Differ. Equ. 50, 289–304 (1983)
Sabinina, E.S.: On a class of non-linear degenerate parabolic equations. Dokl. Akad. Nauk SSSR 143, 794–797 (1962)
Savaré, G., Vespri, V.: The asymptotic profile of solutions of a class of doubly nonlinear equations. Nonlinear Anal. 22, 1553–1565 (1994)
Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and Its Applications, vol. 33. Oxford University Press, Oxford (2006)
Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)
Acknowledgements
This work is partially supported by KAKENHI #22740093 and Hyogo Science and Technology Association.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Italia
About this chapter
Cite this chapter
Akagi, G. (2013). Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_1
Download citation
DOI: https://doi.org/10.1007/978-88-470-2841-8_1
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2840-1
Online ISBN: 978-88-470-2841-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)