Transition from decay to blow-up in a parabolic system

Pavol Quittner

Archivum Mathematicum (1998)

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

Abstract

top
We show a locally uniform bound for global nonnegative solutions of the system u t = Δ u + u v - b u , v t = Δ v + a u in ( 0 , + ) × Ω , u = v = 0 on ( 0 , + ) × Ω , where a > 0 , b 0 and Ω is a bounded domain in n , n 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

top

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

top
  1. H. Brézis, R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differ. Equations, 2 (1977), 601–614 (1977) Zbl0358.35032MR0509489
  2. M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differ. Equations, 98 (1992), 226–240 (1992) Zbl0764.35010MR1170469
  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 (1997) Zbl0892.35069MR1489604
  4. Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415–421 (1986) Zbl0595.35057MR0832917
  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 (1997) Zbl0874.35057MR1423231
  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 (1996) Zbl0858.35059MR1410888
  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) Zbl0919.35057MR1628115
  8. P. Quittner, Signed solutions for a semilinear elliptic problem, Differential and Integral Equations, (to appear) Zbl1131.35339MR1666269

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.