Scaling Variables and Asymptotic Expansions in Damped Wave Equations

Journal of Differential Equations

Volumn 150, Issue 1, 1998, Pages 42-97

  Gallay, Th ^a   Raugel, G ^a  

^a CNRS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001772419     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1998.3459     Document Type: Article

References (48)

1. Babin A. V., Vishik M. I. Attractors of Evolutionary Equations. 1989;North-Holland, Amsterdam.
2. Ball J. M. Strongly continuous semigroups, weak solutions and the variation of constants formula. Proc. Amer. Math. Soc. 63:1977;370-373.
3. Bricmont J., Kupiainen A. Universality in blow-up for non linear heat equations. Nonlinearity. 7:1994;539-575.
4. Bricmont J., Kupiainen A. Stable non-Gaussian diffusive profiles. Nonlinear Anal. 26:1996;583-593.
5. Bricmont J., Kupiainen A., Lin G. Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Comm. Pure Appl. Math. 47:1994;893-922.
6. Cazenave Th., Haraux A. Introduction aux Problèmes d'Evolution Semi-linéaires. Mathématiques et Applications. 1:1990;Ellipses.
7. Crow J. F., Kimura M. An Introduction to Population Genetics Theory. 1969;Harper & Row, New York.
8. Dunbar S. R., Othmer H. G. On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks. Nonlinear Oscillations in Biology and Chemistry. Lecture Notes in Biomathematics. 66:1986;Springer-Verlag, New York/Berlin.
9. Escobedo M., Kavian O., Matano H. Large time behavior of solutions of a dissipative semi-linear heat equation. Comm. Partial Differential Equations. 20:1995;1427-1452.
10. Eckmann J.-P., Wayne C. E., Wittwer P. Geometric stability analysis for periodic solutions of the Swift-Hohenberg equation. Comm. Math. Phys. 190:1997;173-211.
11. N. J. Funct. Anal. 100:1991;119-161.
12. Fulks W., Guenther B. Damped wave equations and the heat equation. Czechoslovak Math. J. 21:1971;683-695.
13. Galaktionov V. A., Kudyumov S. P., Samarskii A. A. On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation. Math. USSR-Sb. 54:1986;421-455.
14. Galaktionov V. A., Vazquez J. L. Asymptotic behaviour of nonlinear parabolic equations with critical exponents: A dynamical system approach. J. Funct. Anal. 100:1991;435-462.
15. Gallay Th., Raugel G. Stability of travelling waves for a damped hyperbolic equation. Z. Angew. Math. Phys. 48:1997;451-479.
16. Th. Gallay, G. Raugel, Scaling variables and stability of hyperbolic fronts.
17. Giga Y., Kohn R. V. Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36:1987;1-40.
18. Goldstein S. On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4:1951;129-156.
19. Hadeler K. P. Reaction telegraph equations and random walk systems. van Strien S., Verduyn Lunel S. Stochastic and Spatial Structures of Dynamical Systems. 1996;North HollandRoyal Academy of the Netherlands, Amsterdam.
20. Hale J. K., Raugel G. Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. J. Differential Equations. 73:1988;197-214.
21. Hale J. K., Raugel G. Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation. J. Dynamics Differential Equations. 2:1990;19-67.
22. Hale J. K., Raugel G. A damped hyperbolic equation on thin domains. Trans. Amer. Math. Soc. 329:1992;185-219.
23. Hardy G. H., Littlewood J. E., Polya G. Inequalities. 1934;Cambridge Univ. Press, Cambridge.
24. Herrero M. A., Velazquez J. J. L. Blow-up profiles in one-dimensional, semilinear parabolic equations. Comm. Partial Differential Equations. 17:1992;205-219.
25. Herrero M. A., Velazquez J. J. L. Blow-up behavior of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré 10:1993;131-189.
26. Hsiao L., Liu T. Convergence to nonlinear diffusion waves for solution of a system of hyperbolic conservation laws with damping. Comm. Math. Phys. 143:1992;599-605.
27. Hsiao L., Liu T. Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chinese Ann. Math. Ser. B. 14:1993;465-480.
28. Kac M. A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4:1974;497-509.
29. Kavian O. Remarks on the large time behavior of a nonlinear diffusion equation. Ann. Inst. H. Poincaré 4:1987;423-452.
30. Lachenmann S. G., Filatrella G., Doderer T., Fernandez J. C., Huebener R. P. Soliton dynamics in two-dimensional Josephson tunnel junctions. Phys. Rev. B. 48:1993;16,623-16,629.
31. Laub A., Doderer T., Lachenmann S. G., Huebener R. P. Lorentz contraction of flux quanta observed in experiments with annular Josephson tunnel junctions. Phys. Rev. Lett. 75:1995;1372-1375.
32. Merle F., Zaag H. Optimal estimates for blow-up rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math. 51:1998;139-196.
33. Mora X., Solà-Morales J. The singular limit of semilinear damped equations. J. Differential Equations. 78:1989;262-307.
34. Murray J. D. Mathematical Biology. Biomathematics. 19:1993;Springer-Verlag, New York/Berlin.
35. Nishihara K. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J. Differential Equations. 131:1996;171-188.
36. Nishihara K. Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J. Differential Equations. 137:1997;384-395.
37. Okubo A. Diffusion and Ecological Problems: Mathematical Models. Biomathematics. 10:1980;Springer-Verlag, New York/Berlin.
38. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci. 44:1983;Springer-Verlag, New York.
39. Raugel G. Dynamics of partial differential equations on thin domains. Johnson R. Dynamical Systems. Lecture Notes in Mathematics. 1609:1995;Springer-Verlag, New York/Berlin.
40. Schneider G. Diffusive stability of spatial periodic solutions of the Swift- Hohenberg equation. Comm. Math. Phys. 178:1996;679-702.
41. Taylor G. I. Diffusion by continuous movements. Proc. London Math. Soc. 20:1920;196-212.
42. Wayne C. E. Invariant manifolds for parabolic partial differential equations in unbounded domains. Arch. Rational Mech. Anal. 138:1997;279-306.
43. Yanagida E. Existence of stable stationary solutions of scalar reaction diffusion equations in thin tubular domains. Appl. Anal. 36:1990;171-188.
44. Zauderer E. Partial Differential Equations of Applied Mathematics. 1983;Wiley, New York.
45. Zlámal M. Sur l'équation des télégraphistes avec un petit paramètre. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 27:1959;324-332.
46. Zlámal M. The mixed problem for hyperbolic equations with a small parameter. Czechoslovak Math. J. 10:1960;83-122.
47. Zlámal M. The parabolic equations as a limiting case of hyperbolic and elliptic equations. Differential Equations and Their Applications, Proc. Conf., Prague. 1963;Academic Press, New York.
48. Zuazua E. A dynamical system approach to the self-similar large time behavior in scalar convection-diffusion equations. J. Differential Equations. 108:1994;1-35.