Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

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.