| Title:
|
A priori estimates of solutions of superlinear problems (English) |
| Author:
|
Quittner, Pavol |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
483-492 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this survey we consider superlinear parabolic problems which possess both blowing-up and global solutions and we study a priori estimates of global solutions. (English) |
| Keyword:
|
a priori estimate |
| Keyword:
|
global existence |
| Keyword:
|
parabolic equation |
| Keyword:
|
superlinear nonlinearity |
| Keyword:
|
blowing-up |
| MSC:
|
35B45 |
| MSC:
|
35J65 |
| MSC:
|
35K60 |
| MSC:
|
35K65 |
| idZBL:
|
Zbl 0977.35029 |
| idMR:
|
MR1844285 |
| DOI:
|
10.21136/MB.2001.134030 |
| . |
| Date available:
|
2009-09-24T21:53:03Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134030 |
| . |
| Reference:
|
[1] H. Brezis, R. E. L. Turner: On a class of superlinear elliptic problems.Commun. Partial Differ. Equations 2 (1977), 601–614. MR 0509489, 10.1080/03605307708820041 |
| Reference:
|
[2] T. Cazenave, P.-L. Lions: Solutions globales d’équations de la chaleur semi linéaires.Commun. Partial Differ. Equations 9 (1984), 955–978. MR 0755928, 10.1080/03605308408820353 |
| Reference:
|
[3] Ph. Clément, D. G. de Figueiredo, E. Mitidieri: A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities.Nonlinear partial differential equations, A. Benkirane at al (eds.), Pitman Research Notes in Math. 343, Harlow, Longman, 1996, pp. 73–91. MR 1417272 |
| Reference:
|
[4] M. Escobedo, M. A. Herrero: Boundedness and blow up for a semilinear reaction-diffusion system.J. Differ. Equations 89 (1991), 176–202. MR 1088342, 10.1016/0022-0396(91)90118-S |
| Reference:
|
[5] M. Fila, P. Souplet, F. Weissler: Linear and nonlinear heat equations in $L^p_\delta $ spaces and universal bounds for global solutions.Preprint. MR 1835063 |
| Reference:
|
[6] V. Galaktionov, J. L. Vázquez: Continuation of blow-up solutions of nonlinear heat equations in several space dimensions.Commun. Pure Applied Math. 50 (1997), 1–67. MR 1423231, 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H |
| Reference:
|
[7] B. Gidas, J. Spruck: A priori bounds for positive solutions of nonlinear elliptic equations.Commun. Partial Differ. Equations 6 (1991), 883–901. MR 0619749 |
| Reference:
|
[8] Y. Giga: A bound for global solutions of semilinear heat equations.Commun. Math. Phys. 103 (1986), 415–421. Zbl 0595.35057, MR 0832917, 10.1007/BF01211756 |
| Reference:
|
[9] Y. Giga, R. V. Kohn: Characterizing blowup using similarity variables.Indiana Univ. Math. J. 36 (1987), 1–40. MR 0876989, 10.1512/iumj.1987.36.36001 |
| Reference:
|
[10] Y. Gu, M. Wang: Existence of positive stationary solutions and threshold results for a reaction-diffusion system.J. Differ. Equations 130 (1996), 277–291. MR 1410888, 10.1006/jdeq.1996.0143 |
| Reference:
|
[11] B. Hu: Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition.Differ. Integral Equations 9 (1996), 891–901. MR 1392086 |
| Reference:
|
[12] S. Kaplan: On the growth of solutions of quasi-linear parabolic equations.Commun. Pure Appl. Math. 16 (1963), 305–330. Zbl 0156.33503, MR 0160044, 10.1002/cpa.3160160307 |
| Reference:
|
[13] H. A. Levine: A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations.Z. Angew. Math. Phys. 42 (1992), 408–430. MR 1115199 |
| Reference:
|
[14] W.-M. Ni, P. E. Sacks, J. Tavantzis: On the asymptotic behavior of solutions of certain quasilinear parabolic equations.J. Differ. Equations 54 (1984), 97–120. MR 0756548, 10.1016/0022-0396(84)90145-1 |
| Reference:
|
[15] P. Quittner: A priori bounds for global solutions of a semilinear parabolic problem.Acta Math. Univ. Comenianae 68 (1999), 195–203. Zbl 0940.35112, MR 1757788 |
| Reference:
|
[16] P. Quittner: Universal bound for global positive solutions of a superlinear parabolic problem.Preprint. Zbl 0981.35010, MR 1839765 |
| Reference:
|
[17] P. Quittner: Signed solutions for a semilinear elliptic problem.Differ. Integral Equations 11 (1998), 551–559. Zbl 1131.35339, MR 1666269 |
| Reference:
|
[18] P. Quittner: A priori estimates of global solutions and multiple equilibria of a parabolic problem involving measure.Preprint. |
| Reference:
|
[19] P. Quittner: Transition from decay to blow-up in a parabolic system.Arch. Math. (Brno) 34 (1998), 199–206. Zbl 0911.35062, MR 1629705 |
| Reference:
|
[20] P. Quittner, Ph. Souplet: In preparation.. |
| Reference:
|
[21] H. Zou: Existence of positive solutions of semilinear elliptic systems without variational structure.Preprint. |
| . |
