Critical global asymptotics in higher-order semilinear parabolic equations

International Journal of Mathematics and Mathematical Sciences

Volumn 2003, Issue 60, 2003, Pages 3809-3825

  Galaktioimov, Victor A ^a  

^a NONE

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EID: 2442713535     PISSN: 01611712     EISSN: None     Source Type: Journal    
DOI: 10.1155/S0161171203210176     Document Type: Article

References (33)

1. F. Bernis, J. Hulshof, and J. L. Vézquez, A very singular solution for the dual porous medium equation and the asymptotic behaviour of general solutions, J. Reine Angew. Math. 435 (1993), 1-31. MathSciNet. Zentralblatt für Mathematik.
2. M. S. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hubert Space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing, Dordrecht, 1987. MathSciNet. Zentralblatt für Mathematik.
3. H. Brezls, L. A. Peletier, and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal. 95 (1986), no. 3, 185-209. MathSciNet. Zentralblatt für Mathematik.
4. J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles, Nonlinear Anal. 26 (1996), no.3, 583-593. CrossRef. MathSciNet. Zentralblatt für Mathematik.
5. J. Bricmont, A. Kupiainen, and G. Lin, Renormalization-group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47 (1994), no. 6, 893-922. MathSciNet. Zentralblatt für Mathematik.
6. M. Chaves and V. A. Galaktionov, Regional blow-up for a higher-order semilinear parabolic equation, European J. Appl. Math. 12 (2001), no. 5, 601-623. MathSciNet. Zentralblatt für Mathematik.
7. S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., Ser. A: Theory Methods 43 (2001), no. 3, 293-323. MathSciNet. Zentralblatt für Mathematik.
8. Ju. L. Dalec'kiǐ and M. G. Kreǐn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, vol. 43, American Mathematical Society, Rhode Island, 1974. MathSciNet. Zentralblatt für Mathematik.
9. Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, and S. I. Pohozaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 2, 93-98. CrossRef. MathSciNet. Zentralblatt für Mathematik.
10. Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, and S. I. Pohozaev, On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range, C. R. Math. Acad. Sci. Paris 335 (2002), no. 10, 805-810. CrossRef. MathSciNet.
11. S. D. Èǐdel'man, Parabolic Systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing, Amsterdam, 1969. MathSciNet. Zentralblatt für Mathematik.
12. A. Friedman, Partial Differential Equations, Robert E. Krieger Publishing, Florida, 1983.
13. V. A. Galaktionov, On a spectrum of blow-up patterns for a higher-order semilinear parabolic equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2011, 1623-1643. CrossRef. MathSciNet. Zentralblatt für Mathematik.
14. V. A. Galaktionov and P. Harwin, Spectra of critical exponents in nonlinear heat equations with absorption, submitted to Int. J. Free Boundaries.
15. V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskii, On asymptotic stability of self-similar solutions of the heat equation with a nonlinear sink, Soviet Math. Dokl. 31 (1985), 271-276. Zentralblatt für Mathematik.
16. V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskii, Asymptotic "eigenfunctions" of the Cauchy problem for a nonlinear parabolic equation, Math. USSR Sbornik 54 (1985), 421-455. Zentralblatt für Mathematik.
17. V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal. 34 (1998), no. 7, 1005-1027. CrossRef. MathSciNet.
18. V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J. 51 (2002), no. 6, 1321-1338. MathSciNet.
19. V. A. Galaktionov and S. A. Posashkov, An approximate self-similar solution of a nonlinear equation of heat conduction with absorption (Moscow, 1984), Mathematical Modeling, Nauka, Moscow, 1989, pp. 103-122 (Russian). MathSciNet.
20. V. A. Galaktionov, S. A. Posashkov, and J. L. Vázquez, Asymptotic convergence to dipole solutions in nonlinear parabolic equations, roc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 5, 877-900. MathSciNet. Zentralblatt für Mathematik.
21. V. A. Galaktionov and J. L. Vázquez, Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical system approach, J. Funct. Anal. 100 (1991), no. 2, 435-462. MathSciNet. Zentralblatt für Mathematik.
22. V. A. Galaktionov and J. F. Williams, On very singular similarity solutions of a higher-order semilinear parabolic equation, to appear in Anal, and Appl.
23. A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in RN , J. Differential Equations 53 (1984), no. 2, 258-276. MathSciNet. Zentralblatt für Mathematik.
24. I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators. Vol. I, Operator Theory: Advances and Applications, vol. 49, Birkhäuser Verlag, Basel, 1990. MathSciNet. Zentralblatt für Mathematik.
25. S. Kamin and M. Ughi, On the behaviour as t→∞ of the solutions of the Cauchy problem for certain nonlinear parabolic equations, J. Math. Anal. Appl. 128 (1987), no. 2, 456-469. MathSciNet. Zentralblatt für Mathematik.
26. S. Kamin and L. Véron, Existence and uniqueness of the very singular solution of the porous media equation with absorption, J. Analyse Math. 51 (1988), 245-258. MathSciNet. Zentralblatt für Mathematik.
27. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Izdat. Nauka, Moscow, 1976. MathSciNet.
28. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MathSciNet. Zentralblatt für Mathematik.
29. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MathSciNet. Zentralblatt für Mathematik.
30. L. A. Peletier and W. C. Troy, Spatial Patterns. Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and Their Applications, vol. 45, Birkhäuser Boston, Massachusetts, 2001. MathSciNet.
31. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, de Gruyter Expositions in Mathematics, vol. 19, Walter de Gruyter, New york, 1995. MathSciNet.
32. G. R. Sell and Y. C. You, Inertial manifolds: the nonselfadjoint case, J. Differential Equations 96 (1992), no. 2, 203-255. MathSciNet. Zentralblatt für Mathematik.
33. M. E. Taylor, Partial Differential Equations. III. Nonlinear Equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. MathSciNet. Zentralblatt für Mathematik.