# Transition from decay to blow-up in a parabolic system

Archivum Mathematicum (1998)

• Volume: 034, Issue: 1, page 199-206
• ISSN: 0044-8753

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## Abstract

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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 $\left(0,+\infty \right)×\Omega$, $u=v=0$ on $\left(0,+\infty \right)×\partial \Omega$, where $a>0$, $b\ge 0$ and $\Omega$ is a bounded domain in ${ℝ}^{n}$, $n\le 2$. In particular, the trajectories starting on the boundary of the domain of attraction of the zero solution are global and bounded.

## How to cite

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Quittner, Pavol. "Transition from decay to blow-up in a parabolic system." Archivum Mathematicum 034.1 (1998): 199-206. <http://eudml.org/doc/18528>.

@article{Quittner1998,
abstract = {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.},
author = {Quittner, Pavol},
journal = {Archivum Mathematicum},
keywords = {Blow-up; global existence; apriori estimates; domain of attraction of the zero solution; global existence; a priori estimates},
language = {eng},
number = {1},
pages = {199-206},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Transition from decay to blow-up in a parabolic system},
url = {http://eudml.org/doc/18528},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Quittner, Pavol
TI - Transition from decay to blow-up in a parabolic system
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 1
SP - 199
EP - 206
AB - 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.
LA - eng
KW - Blow-up; global existence; apriori estimates; domain of attraction of the zero solution; global existence; a priori estimates
UR - http://eudml.org/doc/18528
ER -

## References

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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) (1997) MR1628115
8. P. Quittner, Signed solutions for a semilinear elliptic problem, Differential and Integral Equations, (to appear) Zbl1131.35339MR1666269

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