Previous |  Up |  Next


Blow-up; global existence; apriori estimates
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.
[1] H. Brézis, R. E. L. Turner: On a class of superlinear elliptic problems. Comm. Partial Differ. Equations, 2 (1977), 601–614 MR 0509489
[2] M. Fila: Boundedness of global solutions of nonlinear diffusion equations. J. Differ. Equations, 98 (1992), 226–240 MR 1170469 | Zbl 0764.35010
[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
[4] Y. Giga: A bound for global solutions of semilinear heat equations. Comm. Math. Phys., 103 (1986), 415–421 MR 0832917 | Zbl 0595.35057
[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
[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 MR 1410888 | Zbl 0858.35059
[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
[8] P. Quittner: Signed solutions for a semilinear elliptic problem. Differential and Integral Equations, (to appear) MR 1666269 | Zbl 1131.35339
Partner of
EuDML logo