LIOUVILLE THEOREMS AND BLOW-UP BEHAVIOR IN SEMILINEAR REACTION-DIFFUSION SYSTEMS

This paper is concerned with positive solutions of the semilinear syst em: (S) {u(t) = Delta u + u(p), p greater than or equal to 1, u(t) = D elta u + u(q), q greater than or equal to 1, which blow up at x = 0 an d t = T < infinity. We shall obtain here conditions on p, q and the sp ace dimension N which yield the following bounds on the blow up rates: (1) u(x, t) less than or equal to C(T - t)(-p + 1/pq - 1), u(x, t) le ss than or equal to C(T - t)(-q + 1/pq - 1), for some constant C > 0, We then use (1) to derive a complete classification of blow up pattern s. This last result is achieved by means of a parabolic Liouville theo rem which we retain to be of some independent interest. Finally, we pr ove the existence of solutions of (S) exhibiting a type of asymptotics near blow up which is qualitatively different from those that hold fo r the scalar case.