Blow-up behaviour of one-dimensional semilinear parabolic equations

 Access to full text

 Full (PDF)

1. [A] S. Angenent, The Zero Set of a Solution of a Parabolic Equation, J. reine angew Math., Vol. 390, 1988, pp. 79-96. Zbl0644.35050MR953678
2. [AF] S.B. Angenent and B. Fiedler, The Dynamics of Rotating Waves in Scalar Reaction-Diffusion Equations, Trans. Amer. Math. Soc., Vol. 307, 1988, pp. 545-568. Zbl0696.35086MR940217
3. [AW] D.G. Aronson and H.F. Weinberger, Multidimensional Nonlinear Diffusion arising in Population Genetics, Advances in Math., Vol. 30, 1978, pp. 33-76. Zbl0407.92014MR511740
4. [BBE] J. Bebernes, A. Bressan and D. Eberly, A Description of Blow-up for the Solid Fuel Ignition Model, Indiana Univ. Math. J., Vol. 36, 1987, pp. 131-136. 
5. [B] A. Bressan, On the Asymptotic Shape of Blow-up, Indiana Univ. Math. J., Vol. 39, 1990, pp. 947-960. Zbl0798.35020MR1087180
6. [CMV] X.Y. Chen, H. Matano and L. Veron, Anisotropic Singularities of Solutions of Nonlinear Elliptic Equations in R2, J. Funct. Anal., Vol. 83, 1989, pp. 50-93. Zbl0687.35020MR993442
7. [CL] P.J. Cohen and M. Lees, Asymptotic decay of Differential Inequalities, Pacific J. Math, Vol. 11, 1961, pp. 1235-1249. Zbl0171.35002MR133601
8. [D] J. Dold, Analysis of the Early Stage of Thermal Runaway, Quart. J. Mech. Appl. Math., Vol. 38, 1985, pp. 361-387. Zbl0569.76079
9. [FH] A. Friedman and J.B. McLeod, Blow-up of positive Solutions of Semilinear Heat Equations, Indiana Univ. Math. J., Vol. 34, 1985, pp. 425-447. Zbl0576.35068MR783924
10. [Fu] H. Fujita, On the Blowing-up of Solutions of the Cauchy Problem for ut = Δu + u1+α, J. Fac. Sci. Univ. of Tokio, Section I, Vol. 13, 1966, pp. 109-124. Zbl0163.34002MR214914
11. [GHV] V.A. Galaktionov, M.A. Herrero and J.J.L. Velázquez, The Space Structure near a Blow-up Point for Semilinear Heat Equations: a formal Approach, Soviet J. Comput. Math. and Math. Physics, Vol. 31, 1991, pp. 399-411. Zbl0747.35014MR1107061
12. [GP] V.A. Galakationov and S.A. Posashkov, Application of new Comparison Theorems in the Investigation of Unbounded Solutions of nonlinear Parabolic Equations, Diff. Urav., Vol. 22, 7, 1986, pp. 1165-1173. Zbl0632.35028MR853803
13. [GK1] Y. Giga and R.V. Kohn, Asymptotically Self-Similar Blow-up of Semilinear Heat Equations, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 297-319. Zbl0585.35051MR784476
14. [GK2] Y. Giga and R.V. Kohn, Characterizing Blow-up using Similarity Variables, Indiana Univ. Math., J., Vol. 36, 1987, pp. 1-40. Zbl0601.35052MR876989
15. [GK3] Y. Giga and R.V. Kohn, Nondegeneracy of Blow-up for Semilinear Heat Equations, Comm. Pure Appl. Math., Vol. 42, 1989, pp. 845-884. Zbl0703.35020MR1003437
16. [H] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics No. 840, 1981. Zbl0456.35001MR610244
17. [HV1] M.A. Herrero and J.J.L. Velázquez, Flat Blow-up in One-Dimensional Semilinear Heat Equations, Differential and Integral Equations, Vol. 5, 1992, pp. 973-997. Zbl0767.35036MR1171974
18. [L] A. A. LACEY, The Form of Blow-up for Nonlinear Parabolic Equations, Proc. Royal Soc. Edinburgh, Vol. 98 A, 1984, pp. 183-202. Zbl0556.35077MR765494
19. [La] P.D. Lax, A Stability Theorem for Solutions of Abstract Differential Equations, and its Application to the Study of the Local behaviour of Solutions of Elliptic Equations, Comm. Pure Appl. Math, Vol. 9, 1956, pp. 747-766. Zbl0072.33004MR86991
20. [Li] W. Liu, The Blow-up Rate of Solutions of Semilinear Heat Equations, J. Diff. Equations Vol. 77, 1989, pp. 104-122. Zbl0672.35035MR980545
21. [MW] C.E. Müller and F.B. Weissler, Single Point Blow-up for a General Semilinear Heat Equation, Indiana Univ. Math., J., Vol. 34, 1983, pp. 881-913. Zbl0597.35057MR808833
22. [W] F.B. Weissler, Single Point Blow-up of Semilinear Initial Value Problems, J. Diff. Equations, Vol. 55, 1984, pp. 204-224. Zbl0555.35061MR764124
23. [Wa] N.A. Watson, Parabolic Equations on an Infinite Strip, Monographs and Textbooks inPure and Applied Mathematics, Vol. 127, Marcel Dekker, 1988. Zbl0675.35001MR988890
24. [Wi] D.V. Widder, The Heat Equation, Academic Press, New York, 1975. Zbl0322.35041MR466967

1. Hatem Zaag, Blow-up results for vector-valued nonlinear heat equations with no gradient structure
2. Stathis Filippas, Wenxiong Liu, On the blowup of multidimensional semilinear heat equations
3. Hatem Zaag, On the regularity of the blow-up set for semilinear heat equations
4. Luis Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems
5. M. A. Herrero, J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns
6. D. Andreucci, M. A. Herrero, J. J. L. Velázquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems
7. A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions
8. Frank Merle, Hatem Zaag, Estimations uniformes à l’explosion pour les équations de la chaleur non linéaires et applications
9. Juraj Földes, Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems
10. Thierry Cazenave, Solutions self-similaires de l'équation de Schrödinger non-linéaire