Andreucci, D. and Herrero, Miguel A. and Velázquez, J.J. L.
(1997)
*Liouville theorems and blow up behaviour in semilinear reaction diffusion systems.*
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 14
(1).
pp. 1-53.
ISSN 0294-1449

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Official URL: http://www.sciencedirect.com/science/article/pii/S0294144997801485

## Abstract

This paper is concerned with positive solutions of the semilinear system: (S) {u(t) = Δu + v(p), p ≥ 1, v(t) = Δv + u(q), q ≥ 1, which blow up at x = 0 and t = T < ∞. We shall obtain here conditions on p, q and the space dimension N which yield the following bounds on the blow up rates: (1) u(x, t) ≤ C(T - t)(-p + 1/pq - 1), v(x, t) ≤ C(T - t)(-q + 1/pq - 1), for some constant C > 0. We then use (1) to derive a complete classification of blow up patterns. This last result is achieved by means of a parabolic Liouville theorem which we retain to be of some independent interest. Finally, we prove the existence of solutions of (S) exhibiting a type of asymptotics near blow up which is qualitatively different from those that hold for the scalar case.

Item Type: | Article |
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Uncontrolled Keywords: | Semilinear systems; reaction diffusion equations; asymptotic behaviour; Liouville theorems; a priori estimates; parabolic equations; heat-equations |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 17088 |

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