Backward Blow-up Estimates and Initial Trace for a Parabolic System of Reaction-Diffusion

Abstract

In this article we study the positive solutions of the parabolic semilinear system of competitive type

{u(t) - Delta u + v(p) = 0, u(t) - Delta v + u(q) = 0,

in Omega x (0, T), where Omega is a domain of R(N), and p, q > 0, pq not equal 1. Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case pq > 1 of the form

u(x, t) <= Ct(-(p+1)/(pq-1)), v(x, t) <= Ct(-(q+1)/(pq-1))

in omega x (0, T(1)), for any domain omega subset of subset of Omega, T(1) is an element of (0, T), and C = C(N,p,q,T1,omega). For p, q > 1, we prove the existence of an initial trace at time 0, which is a Borel measure on Omega. Finally we prove that the punctual singularities at time 0 are removable when p,q >= 1+2/N.