Blowup in diffusion equations: A survey

Journal of Computational and Applied Mathematics

Volumn 97, Issue 1-2, 1998, Pages 3-22

  Bandle, Catherine ^a   Brunner, Hermann ^b  

^a UNIVERSITY OF BASEL   (Switzerland)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Criteria for blowup; Finite time blowup; Nonlinear diffusion equations; Numerical solutions

Indexed keywords

ALGORITHMS; CHEMICAL REACTIONS; NUMERICAL ANALYSIS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

DIFFUSION EQUATION; FINITE TIME BLOWUP;

NONLINEAR EQUATIONS;

EID: 0032155992     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(98)00100-9     Document Type: Article

References (92)

1. [1] D.G. Aronson, M.G. Crandall, L.A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982) 1001-1022.
2. [2] C. Bandle, Isoperimetric inequalities for a class of nonlinear parabolic equations, Z. Angew. Math. Phys. 27 (1976) 377-384.
3. [3] C. Bandle, H.A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc. 316 (1989) 595-622.
4. [4] C. Bandle, H.A. Levine, Fujita type phenomena for reaction-diffusion equations with convection like terms, Differential Integral Equations 7 (1994) 1169-1193.
5. [5] C. Bandle, M. Marcus, The positive solutions of a class of semilinear elliptic equations, J. Reine Angew. Math. 401 (1989) 25-59.
6. max for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1985) 142-174.
7. [7] J. Bebernes, D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences 83, Springer, New York, 1989.
8. [8] H. Brézis, T. Cazenave, A nonlinear heat equation with singular initial data, J. Analyse Math. 68 (1996) 277-304.
9. t-Δu = g(u) revisited, Adv. Differential Equations 1 (1996) 73-90.
10. T), Z. Anal. Anwendungen 5 (1986) 237-251.
11. [11] T. Cazenave, P.L. Lions, Solutions globales de la chaleur sémilinéaires, Comm. Partial Differential Equations 9 (1984) 955-978.
12. [12] M. Chipot, F.B. Weissler, Some blowup results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal. 20 (1989) 886-907.
13. [13] M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations 102 (1992) 226-240.
14. [14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
15. [15] A. Friedman, Blow-up solutions of nonlinear parabolic equations, in: W.-M. Ni, L.A. Peletier, J. Serrin (Eds.), Nonlinear Diffusion Equations and Their Equilibrium States, vol. 1, Birkhäuser Verlag, Basel, 1988, pp. 301-318.
16. [16] A. Friedman, B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Math. J. 34 (1985) 425-447.
17. 1+χ, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966) 109-124.
18. [18] H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math. 18, Part I, Amer. Math. Soc. (1968) 138-161.
19. [19] V.A. Galaktionov, On conditions for there to be no global solutions of a class of quasilinear parabolic equations, USSR Comput. Math. Math. Phys. 22 (1982) 73-90.
20. χ-uβ in a neighbourhood of a "singular" point, Soviet Math. Dokl. 33 (1986) 840-844.
21. [21] V.A. Galaktionov, J.L. Vázquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997) 1-67.
22. [22] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phy. 68 (1979) 209-243.
23. [23] Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989) 845-884.
24. [24] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad. Ser. A Math. 49 (1973) 503-525.
25. [25] A.S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys 42 (1987) 169-222.
26. [26] S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math. 16 (1963) 305-333.
27. [27] O. Kavian, Remarks on the large behaviour of a nonlinear differential equation, Ann. Inst. H. Poincaré 4 (1987) 423-452.
28. [28] A.A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Royal Soc. Edinburgh Sect. A 98 (1984) 203-214.
29. [29] A.A. Lacey, D. Tzanetis, Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition, IMA J. Appl. Math. 41 (1988) 207-215.
30. [30] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
31. t = - Au + ℱ (u), Arch. Rat. Mech. Anal. 51 (1973) 371-386.
32. [32] H.A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990) 262-288.
33. [33] P. Meier, Explosion von Lösungen semilinearer parabolischer Differential gleichungen, Dissertation, Universität Basel, 1987.
34. [34] P. Meier, Blow-up solutions of semilinear parabolic equations, Z. Angew. Math. Phys. 39 (1988) 135-149.
35. [35] P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rat. Mech. Anal. 109 (1990) 63-71.
36. [36] K. Mochizuki, R. Suzuki, Critical exponent and critical blow-up for quasilinear parabolic equations, Israel J. Math. 98 (1997) 141-156.
37. [37] P. de Mottoni, A. Schiaffino, A. Tesei, Attractivity properties of nonnegative solutions for a class of nonlinear parabolic problems, Ann. Mat. Pura Appl. 136 (1984) 35-48.
38. [38] W.M. Ni, P.E. Sacks, J. Tavantzis, On the asymptotic behaviour of solutions of certain quasilinear equations of parabolic type, J. Differential Equations 54 (1984) 97-120.
39. [39] S. Ohta, A. Kaneko, Critical exponent of blow-up for semilinear heat equation on a product domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1994) 635-650.
40. [40] L.E. Payne, G.A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives, Math. Methods Appl. Sci. 5 (1995) 95-110.
41. [41] M.H. Protter, H.F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.
42. [42] P. Souplet, F.B. Weissler, Self-similar subsolutions and blow-up for nonlinear parabolic equations, to appear.
43. [43] Y.-W. Qi, The critical exponents of parabolic equations and blow-up in R″, to appear.
44. [44] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995.
45. [45] D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes Math., vol. 309, Springer, New York, 1973.
46. [46] R. Suzuki, Critical blow-up for quasilinear parabolic equations in exterior domains, Tokyo J. Math., to appear.
47. [47] Xuefeng Wang, On Cauchy problems of reaction-diffusion equations, Tokyo J. Math. 19 (1996) 397-409.
48. [48] F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979) 277-296.
49. [49] L.M. Abia, J.C. López-Marcos, J. Martínez, Blow-up for semidiscretizations of reaction-diffusion equations, Appl. Numer. Math. 20 (1996) 145-156.
50. [50] C. Bandle, H. Brunner, Numerical analysis of semilinear parabolic problems with blow-up solutions, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 88 (1994) 203-222.
51. [51] M. Berger, R.V. Kohn, A rescaling algorithm for the numerical calculation of blowing up solutions, Comm. Pure Appl. Math. 41 (1988) 841-863.
52. [52] J.L. Bona, V.A. Dougalis, O.A. Karakashian, W.R. McKinney, Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries-Burger equations, Appl. Numer. Math. 10 (1992) 335-355.
53. [53] C.J. Budd, J. Chen, W. Huang, R.D. Russell, Moving mesh methods with applications to blow-up problems for partial differential equations, in: D.F. Griffiths, G.A. Watson (Eds.), Numerical Analysis 1995 (Dundee 1995), Pitman Res. Notes Math. Ser. 344, Longman, Harlow, 1995, pp. 1-18.
54. [54] C.J. Budd, W. Huang, R.D. Russell, Moving mesh methods for problems with blow-up, SIAM J. Sci. Comput. 17 (1996) 305-327.
55. 1+α, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986) 541-574.
56. [56] Y.-G. Chen, On Blow-Up Solutions of Semilinear Parabolic Equations; Analytical and Numerical Studies, Ph.D. Thesis, Dept. of Mathematics, University of Tokyo, 1987, pp. 65-110.
57. [57] S.N. Dimova, V.A. Galaktionov, D.I. Ivanova, Numerical analysis of blow-up and degeneracy of a semilinear heat equation, Report E11-89-785, Lab. of Computing Techniques and Automation, Joint Institute for Nuclear Research, Dubna, 1989.
58. [58] S.N. Dimova, D.I. Ivanova, Finite element method with special mesh refinement for analysis of single point blow-up solutions, Report E11-91-39, Lab. of Computing techniques and Automation, Joint Institute for Nuclear Research, Dubna, 1991.
59. [59] K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Computational Differential Equations, Cambridge University Press, Cambridge, 1996.
60. [60] K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Advanced Computational Differential Equations, to appear.
61. [61] K. Eriksson, C. Johnson, Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J. Numer. Anal. 32 (1995) 1729-1749.
62. [62] V.A. Galaktionov, A.A. Samarskii, Difference solutions of a class of quasilinear parabolic equations. I., U.S.S.R. Comput. Math. and Math. Phys. 23 (3) (1983) 83-91.
63. [63] V.A. Galaktionov, A.A. Samarskii, On difference solutions of a class of quasi-linear parabolic equations, U.S.S.R. Comput. Math. and Math. Phys. 23 (4) (1983) 40-44.
64. [64] W. Huang, Y. Ren, R.D. Russell, Moving mesh partial differential equations (MMPDES) based on the equidistribution principle, SIAM J. Numer. Anal. 31 (1994) 709-730.
65. [65] W. Huang, R.D. Russell, A moving collocation method for solving time dependent partial differential equations, Appl. Numer. Math. 20 (1996) 101-116.
66. [66] W. Huang, R.D. Russell, Analysis of moving mesh partial differential equations with spatial smoothing, SIAM J. Numer. Anal. 34 (1997) 1106-1126.
67. [67] W. Huang, D.M. Sloan, A simple adaptive grid method in two dimensions, SIAM J. Sci. Comput. 15 (1994) 776-797.
68. 2, Appl. Math. Optim. 2 (1976) 337-350.
69. [69] T. Nakagawa, T. Ushijima, Finite element analysis of the semilinear heat equation of blow-up type, in: J.J.H. Miller (Ed.) Topics in Numerical Analysis III, Academic Press, London, 1977, 275-291.
70. n, J. Math. Anal. Appl. 110 (1986) 456-464.
71. [71] J.M. Sanz-Serna, A.M. Stuart, A note on uniform in time error estimates for approximations to reaction-diffusion equations, IMA J. Numer. Anal. 12 (1992) 457-462.
72. [72] K. Stewart, T. Geveci, Numerical experiments with a nonlinear evolution equation which exhibits blow-up, Appl. Numer. Math. 10 (1992) 139-147.
73. [73] A.M. Stuart, M.S. Floater, On the computation of blow-up, Europ. J. Appl. Math. 1 (1990) 47-71.
74. [74] P.L. Sulem, C. Sulem, A. Patera, Numerical simulation of singular solutions to the two-dimensional cubic Schrödinger equations, Comm. Pure Appl. Math. 37 (1984) 755-578.
75. [75] M. Tabata, A finite difference approach to the number of peaks of solutions for semilinear parabolic problems, J. Math. Soc. Japan 32 (1980) 171-192.
76. [76] Y. Tourigny, J.M. Sanz-Serna, The numerical study of blowup with application to a nonlinear Schrödinger equation, J. Comput. Phys. 102 (1992) 407-416.
77. [77] Y. Tourigny, M. Grinfeld, Deciphering singularities by discrete methods, Math. Comp. 62 (1994) 155-169.
78. [78] J.G. Verwer, J.M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations, Computing 33 (1984) 297-313.
79. [79] H. Bellout, Blow-up solutions of parabolic equations with nonlinear memory, J. Differential Equations 70 (1987) 42-68.
80. [80] C.J. Budd, J.W. Dold, A.M. Stuart, Blow-up in a partial differential equation with conserved first integral, SIAM J. Appl. Math. 53 (1993) 718-742.
81. [81] C.J. Budd, J.W. Dold, A.M. Stuart, Blow-up in a system of partial differential equations with a conserved first integral. II. Problems with convection, SIAM J. Appl. Math. 54 (1994) 610-640.
82. [82] J.M. Chadam, A. Peirce, H.-M. Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313-328.
83. [83] J.S. Guo, H.W. Su, The blow-up behavior of the solution of an integro-differential equation, Differential Integral Equations 5 (1990) 1237-1245.
84. [84] H. Hattori, J.H. Lightbourne, Global existence and blowup for a semilinear integral equation, J. Integral Equations Appl. 2 (1990) 529-546.
85. [85] B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Differential Integral Equations 9 (1996) 891-901.
86. [86] S. Kutzer, Parabolische Differentialgleichungen mit nacheilendem Argument, Dissertation, Universität Basel (1992).
87. [87] M.-N. Le Roux, Semidiscretization in time of a fast diffusion equation, J. Math. Anal. Appl. 137 (1989) 354-370.
88. [88] M.-N. Le Roux, Semidiscretization in time of nonlinear parabolic equations with blowup of the solution, SIAM J. Numer. Anal. 31 (1994) 170-195.
89. [89] C.A. Roberts, D.G. Lasseigne, W.E. Olmstead, Volterra equations which model explosion in a diffuse media, J. Integral Equations Appl. 5 (1993) 531-546.
90. [90] C.A. Roberts, W.E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math. 54 (1996) 153-159.
91. [91] P. Souplet, Blow-up in nonlocal reaction-diffusion equations, manuscript (1997).
92. [92] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, manuscript, 1997.