THE CRITICAL EXPONENT OF DEGENERATE PARABOLIC-SYSTEMS

The Cauchy problem u(t) = DELTAu(alpha) + v(p), v(t) = DELTAv(beta) + u(q) is studied, where x is-an-element-of R(N), 0 < t < infinity and alpha, beta, p and q, are positive exponents. It is proved that if p, q greater-than-or-equal-to 1 and 1 < pq < 1 + 2 max(p + beta, q + alpha)/n then every nontrivial non-negative solution is not global in time; whereas pq > 1 + 2 max(p + beta, q + alpha)/n then there exist both positive global solutions and non-global solutions. In addition, the decaying in time of solutions to u(t) = DELTAu(alpha) in R(n) x [0, infinity), an equation which occurs naturally in our study of above systems, is studied and solutions with the fastest decaying in time are constructed.