| Title:
|
Transition from decay to blow-up in a parabolic system (English) |
| Author:
|
Quittner, Pavol |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
199-206 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show a locally uniform bound for global nonnegative solutions of the system $u_t=\Delta u+uv-bu$, $v_t=\Delta v+au$ in $(0,+\infty )\times \Omega $, $u=v=0$ on $(0,+\infty )\times \partial \Omega $, where $a>0$, $b\ge 0$ and $\Omega $ is a bounded domain in $\mathbb {R}^n$, $n\le 2$. In particular, the trajectories starting on the boundary of the domain of attraction of the zero solution are global and bounded. (English) |
| Keyword:
|
Blow-up |
| Keyword:
|
global existence |
| Keyword:
|
apriori estimates |
| MSC:
|
35B40 |
| MSC:
|
35K50 |
| MSC:
|
35K60 |
| idZBL:
|
Zbl 0911.35062 |
| idMR:
|
MR1629705 |
| . |
| Date available:
|
2009-02-17T10:11:13Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107645 |
| . |
| Reference:
|
[1] H. Brézis, R. E. L. Turner: On a class of superlinear elliptic problems.Comm. Partial Differ. Equations, 2 (1977), 601–614 MR 0509489 |
| Reference:
|
[2] M. Fila: Boundedness of global solutions of nonlinear diffusion equations.J. Differ. Equations, 98 (1992), 226–240 Zbl 0764.35010, MR 1170469 |
| Reference:
|
[3] M. Fila, H. Levine: On the boundedness of global solutions of abstract semi-linear parabolic equations.J. Math. Anal. Appl., 216 (1997), 654–666 MR 1489604 |
| Reference:
|
[4] Y. Giga: A bound for global solutions of semilinear heat equations.Comm. Math. Phys., 103 (1986), 415–421 Zbl 0595.35057, MR 0832917 |
| Reference:
|
[5] V. Galaktionov, J. L. Vázquez: Continuation of blow-up solutions of nonlinear heat equations in several space dimensions.Comm. Pure Applied Math., 50 (1997), 1–67 MR 1423231 |
| Reference:
|
[6] T. Gu, M. Wang: Existence of positive stationary solutions and threshold results for a reaction-diffusion system.J. Diff. Equations, 130, (1996), 277–291 Zbl 0858.35059, MR 1410888 |
| Reference:
|
[7] P. Quittner: Global solutions in parabolic blow-up problems with perturbations.Proc. 3rd European Conf. on Elliptic and Parabolic Problems, Pont-à-Mousson 1997, (to appear) MR 1628115 |
| Reference:
|
[8] P. Quittner: Signed solutions for a semilinear elliptic problem.Differential and Integral Equations, (to appear) Zbl 1131.35339, MR 1666269 |
| . |
