Vázquez, Juan Luis:
The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 15 (2004), fasc. n.3-4, p. 281-300, (English)
pdf (292 Kb), djvu (277 Kb). | MR2148886 | Zbl 1162.35392
Sunto
We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.
Referenze Bibliografiche
[1]
M. ANDERSON,
Geometrization of $3$-Manifolds via the Ricci Flow.
Notices of the AMS,
51, 2,
2004, 184-193. |
MR 2026939 |
Zbl 1161.53350[3]
P. BARAS -
L. COHEN,
Complete blow-up after $T_{max}$ for the solution of a semilinear heat equation.
J. Funct. Anal.,
71,
1987, 142-174. |
fulltext (doi) |
MR 879705 |
Zbl 0653.35037[4]
J. BEBERNES -
D. EBERLY,
Mathematical Problems from Combustion Theory.
Applied Mathematics Series,
83,
Springer-Verlag, New York
1989. |
MR 1012946 |
Zbl 0692.35001[6]
J.D. BUCKMASTER -
G.S.S. LUDFORD,
Lectures on Mathematical Combustion.
CBMS-NSF Regional Conf. Series Appl. Math.,
43,
SIAM, Philadelphia
1983. |
fulltext (doi) |
MR 765073 |
Zbl 0574.76001[7]
E. CHASSEIGNE -
J.L. VÁZQUEZ,
Theory of extended solutions for fast diffusion equations in optimal classes of data. Radiation from singularities.
Arch. Ration. Mech. Anal.,
164,
2002, n. 2, 133-187. |
fulltext (doi) |
MR 1929929 |
Zbl 1018.35048[8]
M. FILA -
H. MATANO,
Connecting equilibria by blow-up solutions. In:
International Conference on Differential Equations (Berlin 1999). 2 voll.,
World Sci. Publishing, River Edge, NJ
2000, 741-743. |
MR 1870227 |
Zbl 0969.35082[9]
A. FRIEDMAN,
Remarks on nonlinear parabolic equations. In:
Applications of Nonlinear Partial Differential Equations in Mathematical Physics.
Amer. Math. Soc., Providence, RI,
1965, 3-23. |
MR 186938 |
Zbl 0192.19601[10]
H. FUJITA,
On the blowing-up of solutions of the Cauchy problem for $u_{t} = \triangle u + u^{1+\alpha}$.
J. Fac. Sci. Univ. Tokyo, Sect. IA Math.,
13,
1966, 109-124. |
MR 214914 |
Zbl 0163.34002[11]
H. FUJITA,
On the nonlinear equations $\triangle u + e^{u} = 0$ and $v_{t} = \triangle v + e^{v}$$\triangle u + e^{u} = 0$ and $v_{t} = \triangle v + e^{v}$.
Bull. Amer. Math. Soc.,
75,
1969, 132-135. |
fulltext mini-dml |
MR 239258 |
Zbl 0216.12101[12]
V.A. GALAKTIONOV -
S.A. POSASHKOV,
The equation $u_{t} = u_{xx} + u^{\beta}$. Localization, asymptotic behaviour of unbounded solutions. Keldysh Inst. Appl. Math. Acad. Sci. USSR, Preprint n. 97,
1985 (in Russian). |
MR 832277[13]
V.A. GALAKTIONOV -
S.A. POSASHKOV,
Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations.
Diff. Urav.,
22,
1986, 1165-1173. |
MR 853803 |
Zbl 0632.35028[14]
V.A. GALAKTIONOV -
J.L. VÁZQUEZ,
Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations.
Arch. Rational Mech. Anal.,
129,
1995, 225-244. |
fulltext (doi) |
MR 1328477 |
Zbl 0827.35055[15]
V.A. GALAKTIONOV -
J.L. VÁZQUEZ,
Continuation of blow-up solutions of nonlinear heat equations in several space dimensions.
Comm. Pure Appl. Math.,
50,
1997, 1-67. |
Zbl 0874.35057[16]
V.A. GALAKTIONOV -
J.L. VÁZQUEZ,
Incomplete blow-up and singular interfaces for quasilinear heat equations.
Comm. Partial Differ. Equat.,
22,
1997, 1405-1452. |
fulltext (doi) |
MR 1469577 |
Zbl 1023.35057[17]
V.A. GALAKTIONOV -
J.L. VÁZQUEZ,
The problem of blow-up in nonlinear parabolic equations. In:
C. CONCA -
M. DEL PINO -
P. FELMER -
R. MANÁSEVICH (eds.),
Current Developments in PDE (a special issue: Proceedings of the Summer Course in Temuco, Chile, jan. 1999).
Discrete Contin. Dynam. Systems, A,
8, 2,
2002, 399-433. |
fulltext (doi) |
MR 1897690 |
Zbl 1010.35057[22]
R. HAMILTON,
The formation of singularities in the Ricci flow.
Surveys in Differential Geometry, vol.
2,
International Press,
1955, 7-136. |
MR 1375255 |
Zbl 0867.53030[23]
S. KAPLAN,
On the growth of solutions of quasilinear parabolic equations.
Comm. Pure Appl. Math.,
16,
1963, 305-330. |
MR 160044 |
Zbl 0156.33503[24]
J.L. KAZDAN -
F.W. WARNER,
Curvature functions for compact 2-manifolds.
Annals of Math.,
99,
1974, 14-47. |
MR 343205 |
Zbl 0273.53034[25] A.N. KOLMOGOROV - I.G. PETROVSKII - N.S. PISKUNOV, A study of a diffusion equation coupled with the growth in the amount of a material, and its application to a biological problem. Byull. Moskov. Gos. Univ., Sect. A, 1, n. 6, 1937, 1-26.
[26]
A.A. LACEY -
D. TZANETIS,
Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition.
IMA J. Appl. Math.,
42,
1988, 207-215. |
fulltext (doi) |
MR 984007 |
Zbl 0699.35151[28]
L.A. LEPIN,
Self-similar solutions of a semilinear heat equation.
Mathematical Modelling,
2,
1990, 63-74 (in Russian). |
MR 1059601 |
Zbl 0972.35506[29]
H.A. LEVINE,
Quenching, nonquenching and beyond quenching for solutions of some parabolic equations.
Ann. Mat. Pura Appl.,
155,
1989, 243-290 (also,
Quenching and beyond: a survey of recent results.
Proc. Int. Conf. on Nonlinear Mathematical Problems in Science and Industry (Iwaki, 1992),
GAKUTO Internat. Ser. Math. Sci. and Appl., vol.
II, Tokyo
1993, 501-512). |
fulltext (doi) |
MR 1042837 |
Zbl 0743.35010[32]
W.F. OSGOOD,
Beweis der Existenz einer Lösung der Differentialgleichung $dy/dx = f(x,y)$ ohne Hinzunahme der Cauchy-Lipschitzschen Bedingung.
Monatshefte für Mathematik und Physik (Vienna),
9,
1898, 331-345. |
fulltext (doi) |
MR 1546565 |
Jbk 29.0260.03[33]
I. PERAL -
J.L. VÁZQUEZ,
On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term.
Arch. Rat. Mech. Anal.,
129,
1995, 201-224. |
fulltext (doi) |
MR 1328476 |
Zbl 0821.35080[34]
F. QUIRÓS -
J.D. ROSSI -
J.L. VÁZQUEZ,
Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions.
Comm. Partial Differential Equations,
27,
2002, 395-424. |
fulltext (doi) |
MR 1886965 |
Zbl 0996.35036[35]
F. QUIRÓS -
J.D. ROSSI -
J.L. VÁZQUEZ,
Thermal avalanche for blow-up solutions of semilinear heat equations.
Comm. Pure Appl. Math.,
57,
2004, n. 1, 59-98. |
fulltext (doi) |
MR 2007356 |
Zbl 1036.35084[36]
A.A. SAMARSKII -
V.A. GALAKTIONOV -
S.P. KURDYUMOV -
A.P. MIKHAILOV,
Blow-up in problems for quasilinear parabolic equations.
Nauka, Moscow
1987 (in Russian; English transl.:
Walter de Gruyter, Berlin
1995). |
Zbl 1020.35001[37]
J. SMOLLER,
Shock-Waves and Reaction-Diffusion Equations.
Springer-Verlag, New York
1983. |
MR 688146 |
Zbl 0807.35002[38]
J.L. VÁZQUEZ,
Domain of existence and blowup for the exponential reaction-diffusion equation.
Indiana Univ. Math. J.,
48, 2,
1999, 677-709. |
fulltext (doi) |
MR 1722813 |
Zbl 0928.35080[39]
J.L. VÁZQUEZ,
Asymptotic behaviour for the $PME$ in the whole space.
J. Evol. Equ.,
3,
2003, n. 1, 67-118. |
Zbl 1036.35108[40]
J.J.L. VELÁZQUEZ,
Classification of singularities for blowing-up solutions in higher dimensions.
Trans. Amer. Math. Soc.,
338,
1993, 441-464. |
fulltext (doi) |
MR 1134760 |
Zbl 0803.35015[41]
YA.B. ZEL'DOVICH -
G.I. BARENBLATT -
V.B. LIBROVICH -
G.M. MAKHVILADZE,
The Mathematical Theory of Combustion and Explosions.
Consultants Bureau, New York
1985. |
fulltext (doi) |
MR 781350[42] YA.B. ZEL'DOVICH - D.A. FRANK-KAMENETSKII, The theory of thermal propagation of flames. Zh. Fiz. Khim., 12, 1938, 100-105 (in Russian; English transl. in: Collected Works of Ya.B. Zeldovich, vol. 1, Princeton Univ. Press, 1992).
Con il contributo del Ministero per i Beni e le Attività Culturali