On similarity solutions and blow-up spectra for a semilinear wave equation

Quarterly of Applied Mathematics

Volumn 61, Issue 3, 2003, Pages 583-600

  Galaktionov, V A ^a,b   Pohozaev, S I ^c  

^a University of Bath   (United Kingdom)

^b KELDYSH INSTITUTE OF APPLIED MATHEMATICS   (Russian Federation)

^c STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; FINITE ELEMENT METHOD; LINEARIZATION; MATHEMATICAL OPERATORS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; POLYNOMIALS; PROBLEM SOLVING;

BLOW UP PATTERNS; BLOW-UP SPECTRA; CAUCHY PROBLEM; ELLIPTIC EQUATIONS; FIBERING METHODS; SEMILINEAR WAVE EQUATIONS;

ASYMPTOTIC STABILITY;

EID: 0141885009     PISSN: 0033569X     EISSN: None     Source Type: Journal    
DOI: 10.1090/qam/1999839     Document Type: Article

References (47)

1. S. Alinhac, Blow-up for Nonlinear Hyperbolic Equations, Birkhäuser, Boston/Berlin, 1995
2. D. Amadori, Unstable blow-up patterns, Differ. Integr. Equat., 8, 1977-1996 (1995)
3. H. Bateman and E. Erdélyi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953
4. M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel, Dordrecht/Tokyo, 1987
5. A. Bressan, Stable blow-up patterns, J. Differ. Equat., 98, 57-75 (1992)
6. H. Brezis, L. A. Peletier, and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal., 95, 185-209 (1986)
7. 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)
8. L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 223-241 (1986)
9. Ju. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs, Vol. 43, Amer. Math. Soc., Providence, RI, 1974
10. p, Comm. Pure Appl. Math., 45, 821-869 (1992)
11. 1+α, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math., 13, 109-124 (1966)
12. V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskii, On asymptotic "eigenfunctions" of the Cauchy problem for a nonlinear parabolic equation, Math. USSR Sbornik, 54, 421-455 (1986)
13. V. A. Galaktionov and J. L. Vazquez, Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differ. Equat., 127, 1-40 (1996)
14. V. A. Galaktionov and J. L. Vazquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50, 1-68 (1997)
15. V. Georgiev, H. Lindblad, and C. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119, 1291-1319 (1997)
16. H. P. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J., 33, 573-582 (1984)
17. D. B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differ. Equat., 59, 165-205 (1985)
18. M. A. Herrero and J. J. L. Veláquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, Analyse non linéaire, 10, 131-189 (1993)
19. L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Math. and Appl., Vol. 26, Springer-Verlag, Berlin/New York, 1997
20. F. John, Blow-up of solutions of nonlinear wave equation in three space dimensions, Manuscripta Math., 28, 235-268 (1979)
21. S. Kamin and L. A. Peletier, Large time behaviour of solutions of the heat equation with absorption, Ann. Sc. Norm. Pisa Cl. Sci. (4), 12, 393-408 (1984)
22. S. Kamin and L. A. Peletier, Large time behaviour of solutions of the porous media equation with absorption, Israel J. of Math., 55, 129-146 (1996)
23. T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 32, 501-505 (1980)
24. O. Kavian and F. B. Weissler, Finite energy self-similar solutions of a nonlinear wave equation, Comm. Partial Differ. Equat., 15, 1381-1420 (1990)
25. J. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math., 10, 523-530 (1957)
26. S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, I and II, Comm. Partial Differ. Equat., 18, 431-452, and 1869-1899 (1993)
27. A. N. Komogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1976
28. S. G. Krein, Linear Differential Equations in Banach Space, Transl. Math. Monographs, Vol. 29, Amer. Math. Soc., Providence, RI, 1971
29. B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Self-Adjoint Ordinary Differential Operators, Transl. Math. Mon., Vol. 39, Amer. Math. Soc., Providence, RI, 1975
30. H. Lingblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Differ. Equat., 130, 357-426 (1995)
31. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 17, Amer. Math. Soc. Providence, RI, 1988
32. p-1u, Duke Math. J., 86, 143-195 (1997)
33. E. Mitidieri and S. I. Pohozaev, A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities, Proc. Steklov Math. Inst., 3, Vol. 234, Moscow, 2001 (ISSN:0081-5438)
34. M. A. Naimark, Linear Differential Operators, Part 1, Frederick Ungar Publ. Co., New York, 1967
35. H. Pecher, Sharp existence results for self-similar solutions of semilinear wave equations, Nonl. Differ. Equat. Appl., 7, 323-341 (2000)
36. H. Pecher, Self-similar and asymptotically self-similar solutions of nonlinear wave equations, Math. Ann., 316, 259-281 (2000)
37. F. Planchon, Self-similar solutions and semi-linear wave equations in Besov spaces, J. Math. Anal. Appl., 78, 809-820 (2000)
38. S. I. Pohozaev, Eigenfunctions of the equation Δu + λf(u) = 0, Soviet Math. Dokl., 6, 1408-1411 (1965)
39. S. I. Pohozaev, On an approach to nonlinear equations, Soviet Math. Dokl., 20, 912-916 (1979)
40. S. I. Pohozaev, The fibering method in nonlinear variational problems, Pitman Research Notes in Math., Vol. 365, Pitman, pp. 35-88 (1997)
41. S. I. Pohozaev and L. Véron, Blow-up results for nonlinear hyperbolic inequalities, Annali Scuola Norm. Sup. Pisa, Ser. IV, 29, 393-420 (2000)
42. F. Ribaud and A. Youssfi, Solutions globales et solutions auto-similaires de l'équation des ondes non linéaire, C. R. Acad. Sci. Paris Sér. I Math. 329, 33-36 (1999)
43. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin/New York, 1995
44. T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differ. Equat., 32, 378-406 (1984)
45. C. Sturm, Mémoire sur une classe d'équations à différences partielles, J. Math. Pures Appl., 1, 373-444 (1836)
46. J. J. L. Velázquez, 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)
47. J. J. L. Velázquez, V. A. Galaktionov, and M. A. Herrero, The space structure near a blow-up point for semilinear heat equations: a formal approach, USSR Comput. Math. Phys., 31, 46-55 (1991)