Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations

Royal Society of Edinburgh - Proceedings A

Volumn 133, Issue 5, 2003, Pages 1075-1119

  Galaktionov, V A ^a,b   Shishkov, A E ^c  

^a University of Bath   (United Kingdom)

^b KELDYSH INSTITUTE OF APPLIED MATHEMATICS   (Russian Federation)

^c INSTITUTE OF APPLIED MATHEMATICS AND MECHANICS   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345412729     PISSN: 03082105     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0308210500002821     Document Type: Conference Paper

References (60)

1. V. F. Akulov and A. E. Shishkov. On asymptotic properties of solutions of mixed problems for quasilinear parabolic equations in unbounded domains. Mat. USSR Sb. 73 (1992), 557-567.
2. H. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311-341.
3. G. I. Barenblatt. Scaling, self-similarity, and intermediate asymptotics (Cambridge University Press, 1996).
4. A.-J.-C. Barré de Saint-Venant. De la torsion des prismes (Paris: Imprimére Impériale, 1855).
5. A.-J.-C. Barré de Saint-Venant. Mémoire sur la torsion des prismes. Mémoires Divers des Savants étrangers. Acad. Sci. Paris 14 (1856), 233-560.
6. A.-J.-C. Barré de Saint-Venant. Mémoire sur la flexion des prismes. J. Mathé. Liouville Sér. II 1 (1856), 89.
7. N. G. Basov and O. N. Krohin. Conditions of plasma heating by optic generators radiation. J. Exp. Tech. Phys. 46 (1964), 171-180.
8. J. Bebernes and D. Eberly. Mathematical problems in combustion theory. Applied Mathematical Science, vol. 83 (Springer, 1989).
9. F. Bernis. Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension n. Commun. PDEs 9 (1984), 271-312.
10. F. Bernis. Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption. Proc. R. Soc. Edinb. A 104 (1986), 1-19.
11. F. Bernis. Qualitative properties for some nonlinear higher-order degenerate parabolic equations. Houston J. Math. 14 (1988), 319-352.
12. F. Bernis. Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. Math. Ann. 279 (1988), 373-394.
13. 2, θ (Ω, δ). Annli Mat. Pura Appl. 73 (1966), 55-102.
14. E. DiBenedetto. Degenerate parabolic equations. Series Universitext (Springer, 1993).
15. V. A. Galaktionov and A. A. Samarskii, Methods of construction of approximate self-similar solutions of nonlinear heat equations. IV. Mat. USSR Sb. 49 (1984), 125-149.
16. V. A. Galaktionov and A. E. Shishkov. Structure of boundary blow-up in higher-order quasilinear parabolic equations. (Available at http://www.maths.bath.ac.uk/mathematics/preprints.html.) (Preprint.)
17. V. A. Galaktionov and A. E. Shishkov. Self-similar boundary blow-up structures in higher-order quasilinear parabolic equations. (Available at http://www.maths.bath.ac.uk/mathematics/preprints.html.) (Preprint.)
18. V. A. Galaktionov and J. L. Vazquez. Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation. SIAM J. Math. Analysis 24 (1993), 1254-1276.
19. V. A. Galaktionov and J. L. Vazquez. Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations. J. Diff. Eqns 127 (1996), 1-40.
20. V. A. Galaktionov and J. L. Vazquez. Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50 (1997), 1-68.
21. V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii. Heat localization in nonlinear media. Diff. Eqns 17 (1981), 1141-1154.
22. I. M. Gel'fand. Some problems in the theory of quasilinear equations. Am. Math. Soc. Transl. 2 29 (1963), 295-381.
23. B. H. Gilding and M. A. Herrero. Localization and blow-up of thermal waves in nonlinear heat conduction. Math. Ann. 282 (1988), 223-242.
24. A. S. Kalashnikov. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russ. Math. Surv. 42 (1987), 169-222.
25. J. K. Knowles. On Saint-Venant's principle in the two-dimensional linear theory of elasticity. Arch. Ration. Mech. Analysis 21 (1966), 1-22.
26. J. K. Knowles. A Saint-Venant's principle for a class of second-order elliptic boundary-value problems. Z. Angew. Math. Phys. 18 (1967), 473-490.
27. J. K. Knowles. On the spatial decay of the heat equation. Z. Angew. Math. Phys. 22 (1971), 1050-1056.
28. V. A. Kondratiev and O. A. Oleinik. On the behaviour of generalized solutions of the Dirichlet problem for higher-order elliptic equations in a neighbourhood of the boundary. Zap. Nauchnykh Semin. LOMI 115 (1982), 114-125.
29. S. N. Kruzhkov. First-order quasilinear equations in several independent variables. Mat. USSR Sb. 10 (1970), 217-243.
30. S. P. Kurdyumov. Nonlinear processes in dense plasma. In Proc. 2nd Int. Conf. on Plasma Theory, Kiev, 1974, pp. 278-287 (Kiev: Naukova Dumka, 1976).
31. S. P. Kurdyumov. Evolution and self-organization laws in complex systems. Int. J. Mod. Phys. C 1 (1990), 299-327.
32. S. P. Kurdyumov, A. A. Samarskii and N. V. Zmitrenko. Heat localization effects in problems of ICF (inertial confinement fusion). Int. J. Mod. Phys. B 9 (1995), 1797-1811.
33. H. A. Levine. The role of critical exponents in blow-up problems. SIAM Rev. 32 (1990), 262-288.
34. J. L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires (Paris: Dunod, 1969).
35. H. Nawa. Asymptotic and limiting profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power. Commun. Pure Appl. Math. 52 (1999), 193-270.
36. I. Nuckolls, L. Wood, A. Thiessen and G. Zimmerman. Laser compression of matter to super-high densities. Nature 239 (1972), 139-142.
37. O. A. Oleinik. On the uniqueness of the solutions of the Cauchy problem for general parabolic systems in classes of rapidly increasing functions. Usp. Mat. Nauk 29 (1974), 229-230.
38. O. A. Oleinik and G. A. Iosif'yan. An analogue of Saint-Venant's principle and the uniqueness of solutions of boundary-value problem for parabolic equations in unbounded domains. Russ. Math. Surv. 31 (1976), 153-178.
39. O. A. Oleinik and G. A. Iosif'yan. Boundary value problems for second order elliptic equations in unbounded domains and Saint-Venant's principle. Annli Scuola Norm. Sup. Pisa 2 (1977), 269-290.
40. O. A. Oleinik and G. A. Iosif'yan. On singularities at the boundary points and uniqueness theorems for solutions of the first boundary-value problem of elasticity. Commun. PDEs 2 (1977), 937-969.
41. O. A. Oleinik and E. V. Radkevich. Method of introducing of a parameter for evolution equations. Russ. Math. Surv. 33 (1978), 7-84.
42. L. A. Peletier and W. C. Troy. Spatial patterns. higher order models in physics and mechanics (Birkhäuser, 2001).
43. A. A. Samarskii and I. M. Sobol'. Examples of numerical computation of temperature waves. USSR Comput. Math. Math. Phys. 3 (1963), 945-970.
44. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov. Localization of diffusion processes in media with constant properties. Sov. Phys. Dokl. 24 (1979), 543-545.
45. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov. Blow-up in quasilinear parabolic equations (Berlin: Walter de Gruyter, 1995).
46. A. E. Shishkov. Behaviour of generalized solutions of mixed problems for quasilinear parabolic equations of high order in unbounded domain. Ukrain. Mat. Zh. 39 (1988), 509-515.
47. A. E. Shishkov. Dynamics of the support geometry for generalized solutions to a quasilinear divergent parabolic equation of higher order. Diff. Eqns 29 (1993), 460-469.
48. A. E. Shishkov. Evolution of the support of a solution with unbounded energy of a quasilinear degenerate parabolic equation of arbitrary order. Sb. Math. 186 (1995), 1843-1864.
49. A. E. Shishkov. Boundary blow-up for energy solutions of general multi-dimensional parabolic equations. Nonlin. Bound. Value Prob. 8 (1998), 229-237.
50. A. E. Shishkov and A. G. Shchelkov. Blow-up boundary regimes for general quasilinear parabolic equations in multidimensional domains. Sb. Math. 190 (1999), 447-479.
51. V. A. Solonnikov. On differential properties of weak solutions of quasi-linear elliptic equations. Zap. Nauchnykh Semin. LOMI 30 (1972), 174-175.
52. V. A. Solonnikov. Differential properties of weak solutions of quasi-linear elliptic equations. Zap. Nauchnykh Semin. LOMI 39 (1974), 110-119. (Transl. J. Soviet Math. 8 (1977), 79-86.)
53. V. A. Solonnikov. Differential properties of weak solutions of quasi-linear elliptic equations. Zap. Nauchnykh Semin. LOMI 39 (1974), 110-119. (Transl. J. Soviet Math. 8 (1977), 79-86.)
54. C. Sulem and P.-L. Sulem. The nonlinear Schrödinger equation. Applied Mathematical Science, vol. 139 (Springer, 1999).
55. R. A. Toupin. Saint-Venant's principle. Arch. Ration. Mech. Analysis 18 (1965), 83-96.
56. Ya. B. Zel'dovich and Yu. P. Raizer. Physics of shock waves and high-temperature hydrodynamic phenomena, vol. I (Academic, 1966).
57. Ya. B. Zel'dovich and Yu. P. Raizer. Physics of shock waves and high-temperature hydrodynamic phenomena, vol. II (Academic, 1967).
58. Ya. B. Zel'dovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze. The mathematical theory of combustion and explosions (New York: Plenum, 1985).
59. N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii. Localization of thermonuclear combustion in plasma with electron heat conductance. Pis'ma Zh. Eksper. Tear. Fiz. 26 (1977), 620-624. (Transl. JETP Lett. 26 (1977).)
60. N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii. Localization of thermonuclear combustion in plasma with electron heat conductance. Pis'ma Zh. Eksper. Tear. Fiz. 26 (1977), 620-624. (Transl. JETP Lett. 26 (1977).)