Hermitian spectral theory and blow-up patterns for a fourth-order semilinear boussinesq equation

Studies in Applied Mathematics

Volumn 121, Issue 4, 2008, Pages 395-431

  Galaktionov, V A ^a  

^a University of Bath   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL OPERATORS; POLYNOMIALS; WATER WAVES; WAVE EQUATIONS;

ASYMPTOTICS; BLOW-UP; BOUSSINESQ EQUATIONS; BOUSSINESQ-TYPE EQUATIONS; CAUCHY PROBLEMS; FOURTH-ORDER; HERMITIANS; MATCHINGS; SEMILINEAR; SPECTRAL THEORY;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 55649110251     PISSN: 00222526     EISSN: 14679590     Source Type: Journal    
DOI: 10.1111/j.1467-9590.2008.00421.x     Document Type: Article

References (56)

1. S. Alinhac, Blow-up for Nonlinear Hyperbolic Equations, Birkhäuser, Boston/Berlin, 1995.
2. L. Bers, Local behaviour of solutions of general linear elliptic equations, Comm. Pure Appl. Math. 8 : 473 496 (1955).
3. M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel, Dordrecht/Tokyo, 1987.
4. R. Bizon, D. Maison, and A. Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity 20 : 2061 2074 (2007).
5. C. Budd and V. Galaktionov, Stability and spectra of blow-up in problems with quasi-linear gradient diffusivity, Proc. Roy. Soc. London A 454 : 2371 2407 (1998).
6. L. A. Caffarelli and A. Friedman, Differentiability of the blow-up curve for one dimensional nonlinear wave equations, Arch. Rational Mech. Anal. 91 : 83 98 (1985).
7. L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc. 297 : 223 241 (1986).
8. X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 : 603 630 (1998).
9. X.-Y. Chen, On the scaling limits at zeros of solutions pf parabolic equations, J. Differ. Equat. 147 : 355 382 (1998).
10. E. F. Chladni, Entdeckungen über die Theorie des Klanges, Weidmanns Erben ynd Reich, Leipzig, 1787.
11. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc. New York/London, 1955.
12. R. DiPerna, Measured-valued solutions to conservation laws, Arch. Ration. Mech. Anal. 88 : 223 270 (1985).
13. Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, and S. I. Pohozaev, Asymptotic behaviour of global solutions to higher-order semilinear parabolic equations in the supercritical range, Adv. Differ. Equat. 9 : 1009 1038 (2004).
14. L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J. 104 : 113 127 (2000).
15. J. D. Evans, V. A. Galaktionov, and J. R. King, Source-type solutions of the fourth-order unstable thin film equation, Euro J. Appl. Math. 18 : 273 321 (2007).
16. J. D. Evans, V. A. Galaktionov, and J. R. King, Unstable sixth-order thin film equation. I. Blow-up similarity solutions; II. Global similarity patterns, Nonlinearity 20 : 1799 1841, 1843-1881 (2007).
17. p, Comm. Pure Appl. Math. 45 : 821 869 (1992).
18. V. A. Galaktionov, On a spectrum of blow-up patterns for a higher-order semilinear parabolic equations, Proc. Roy. Soc. London A 457 : 1 21 (2001).
19. V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall/CRC, Boca Raton, Florida, 2004.
20. V. A. Galaktionov, Sturmian nodal set analysis for higher-order parabolic equations and applications, Adv. Differ. Equat. 12 : 669 720 (2007).
21. V. A. Galaktionov, Shock waves and compactons for fifth-order nonlinear dispersion equations, Europ. J. Appl. Math. submitted.
22. V. A. Galaktionov, Formation of shocks and fundamental solution of a fourth-order quasilinear Boussinesq-type equation, Nonlinearity submitted.
23. V. A. Galaktionov and S. I. Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation, Quart. Appl. Math. 61 : 583 600 (2003).
24. V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC, Boca Raton, Florida, 2007.
25. G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 : 314 317 (1920).
26. H. P. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J. 33 : 573 582 (1984).
27. L. M. Hocking, K. Stewartson, and J. T. Stuart, A nonlinear instability burst in plane parallel flow, J. Fluid Mech. 51 : 705 735 (1972).
28. M. Inc, New compacton and solitary pattern solutions of the nonlinear modified dispersive Klein-Gordon equations, Chaos, Solitons and Fractals 33 : 1275 1284 (2007).
29. A. D. Ionescu and C. E. Kenig, Uniqueness properties of solutions of Schrödinger equations, J. Funct. Anal. 232 : 90 236 (2006).
30. D. Jakobson, N. Nadirashvili, and J. Toth, geometric properties of eigenfunctions, Russian Math. Surveys 56 : 1085 1105 (2001).
31. F. John, Blow-up of solutions of nonlinear wave equation in three space dimensions, Manuscripta Math. 28 : 235 268 (1979).
32. T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 32 : 501 505 (1980).
33. O. Kavian and F. B. Weissler, Finite energy self-similar solutions of a nonlinear wave equation, Comm. Partial Differ. Equat. 15 : 1381 1420 (1990).
34. M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 : 955 380 (1998).
35. J. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10 : 523 530 (1957).
36. S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, I, II, Comm. Partial Differ. Equat. 18 : 431 452, 1869-1899 (1993).
37. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1976.
38. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.
39. L. Ljusternik and V. Sobolev, Elements of Functional Analysis, Ungar Publ. Comp. New York, 1961.
40. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon. Vol. 71, Amer. Math. Soc., Providence, RI, 1988.
41. V. G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin/Tokyo, 1985.
42. p-1u Duke Math. J. 86 : 143 195 (1997).
43. F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equations in one space dimension, J. Funct. Anal. 253 : 43 121 (2007).
44. E. Mitidieri and S. I. Pohozaev, A Priori Estimates and Blow-up of Solutions to Nonlinear Partial ifferential Equations and Inequalities, Proc. Steklov Math. Inst. 3, Vol. 234, Moscow, 2001 (ISSN: 0081-5438).
45. F. Murat, Compacité por compensation, Ann. Scuola Norm. Sup. Pisa Sci. Math., 5 : 489-507 (1978);
46. F. Murat, Compacité por compensation, Ann. Scuola Norm. Sup. Pisa Sci. Math. 8 : 69 102 (1981).
47. M. A. Naimark, Linear Differential Operators, Part 1, Frederick Ungar Publ. Co., New York, 1967.
48. S. I. Pohozaev and L. Véron, Blow-up results for nonlinear hyperbolic inequalities, Annali Scuola Norm. Sup. Pisa, Ser. IV 29 : 393 420 (2000).
49. J. Rayleigh, The Theory of Sound, Macmillan, London, 1926 (1st ed., 1877).
50. T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differ. Equat. 32 : 378 406 (1984).
51. C. Sturm, Mémoire sur une classe d'équations à différences partielles, J. Math. Pures Appl. 1 : 373 444 (1836).
52. L. Tartar, Compensated Compactness and Applications to PDEs, Research Notes in Math., Vol. 39, Pitman Publ. Inc., New York, 1979.
53. J. J. L. Velazquez, Estimates on (N - 1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 : 445 476 (1993).
54. J. J. L. Velazquez, V. A. Galaktionov, and M. A. Herrero, The space structure near a blow-up point for semilinear heat equations: A formal approach, Comput. Math. Math. Phys. 31 : 46 55 (1991).
55. M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher-dimension, Duke Math. J. 138 : 281 374 (2007).
56. Z. Yan, Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation II. Solitary pattern solutions, Chaos, Solitons and Fractals 18 : 869 880 (2003).