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Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

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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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
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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