Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Authors: V. A. Galaktionov and S. I. Pohozaev
Journal: Quart. Appl. Math. 61 (2003), 583-600
MSC: Primary 35L70; Secondary 35B40
DOI: https://doi.org/10.1090/qam/1999839
MathSciNet review: MR1999839
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct countable spectra of different asymptotic patterns of self-similar and approximate self-similar types for global and blow-up solutions for the semilinear wave equation

$\displaystyle {u_{tt}} = \Delta u + {\left\vert u \right\vert^{p - 1}}u, \qquad x \in {R^N}, t > 0,$

in different ranges of exponent $ p$ and dimension $ N$.

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  • [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) MR 1348961
  • [3] H. Bateman and E. Erdélyi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953 MR 0058756
  • [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) MR 1168971
  • [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) MR 853963
  • [7] L. A. Caffarelli and A. Friedman, Differentiability of the blow-up curve for one dimensional non-linear wave equations, Arch. Rational Mech. Anal., 91, 83-98 (1985) MR 802832
  • [8] L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 223-241 (1986) MR 849476
  • [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 MR 0352639
  • [10] S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of $ {u_t} - \Delta u = {u^p}$, Comm. Pure Appl. Math., 45, 821-869 (1992) MR 1164066
  • [11] H. Fujita, On the blowing up of solutions to the Cauchy problem for $ {u_t} = \Delta u + {u^{1 + \alpha }}$, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math., 13, 109-124 (1966) MR 0214914
  • [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) MR 788082
  • [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) MR 1387255
  • [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) MR 1423231
  • [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) MR 1481816
  • [16] H. P. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J., 33, 573-582 (1984) MR 749315
  • [17] D. B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differ. Equat., 59, 165-205 (1985) MR 804887
  • [18] M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, Analyse non linéaire, 10, 131-189 (1993) MR 1220032
  • [19] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Math. and Appl., Vol. 26, Springer-Verlag, Berlin/New York, 1997 MR 1466700
  • [20] F. John, Blow-up of solutions of nonlinear wave equation in three space dimensions, Manuscripta Math., 28, 235-268 (1979) MR 535704
  • [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) MR 837255
  • [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 (1986) MR 868174
  • [23] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 32, 501-505 (1980) MR 575735
  • [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) MR 1077471
  • [25] J. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math., 10, 523-530 (1957) MR 0096889
  • [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) MR 1214867
  • [27] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1976 MR 0435771
  • [28] S. G. Krein, Linear Differential Equations in Banach Space, Transl. Math. Monographs, Vol. 29, Amer. Math. Soc., Providence, RI, 1971 MR 0342804
  • [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 MR 0369797
  • [30] H. Lingblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Differ. Equat., 130, 357-426 (1995) MR 1335386
  • [31] A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 71, Amer. Math. Soc., Providence, RI, 1988 MR 971506
  • [32] F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $ {u_t} = \Delta u + {\left\vert u \right\vert^{p - 1}}u$, Duke Math. J., 86, 143-195 (1997) MR 1427848
  • [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) MR 1879326
  • [34] M. A. Naimark, Linear Differential Operators, Part 1, Frederick Ungar Publ. Co., New York, 1967 MR 0216050
  • [35] H. Pecher, Sharp existence results for self-similar solutions of semilinear wave equations, Nonl. Differ. Equat. Appl., 7, 323-341 (2000) MR 1807462
  • [36] H. Pecher, Self-similar and asymptotically self-similar solutions of nonlinear wave equations, Math. Ann., 316, 259-281 (2000) MR 1741271
  • [37] F. Planchon, Self-similar solutions and semi-linear wave equations in Besov spaces, J. Math. Anal. Appl., 78, 809-820 (2000) MR 1782103
  • [38] S. I. Pohozaev, Eigenfunctions of the equation $ \Delta u + \lambda f\left( u \right) = 0$, Soviet Math. Dokl., 6, 1408-1411 (1965) MR 0192184
  • [39] S. I. Pohozaev, On an approach to nonlinear equations, Soviet Math. Dokl., 20, 912-916 (1979) MR 550349
  • [40] S. I. Pohozaev, The fibering method in nonlinear variational problems, Pitman Research Notes in Math., Vol. 365, Pitman, pp. 35-88 (1997) MR 1478747
  • [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) MR 1784180
  • [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) MR 1703283
  • [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 MR 1330922
  • [44] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differ. Equat., 32, 378-406 (1984) MR 744303
  • [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 $ \left( N - 1 \right)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J., 42, 445-476 (1993) MR 1237055
  • [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) MR 1107061

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DOI: https://doi.org/10.1090/qam/1999839
Article copyright: © Copyright 2003 American Mathematical Society

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