GLOBAL EXISTENCE AND BLOW UP FOR SYSTEMS OF NONLINEAR WAVE EQUATIONS RELATED TO THE WEAK NULL CONDITION

We discuss how the higher-order term vertical bar u vertical bar(q) (q > 1+2/(n- 1)) has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations partial derivative(2)(t)u - Delta u = vertical bar v vertical bar(p), partial derivative(2)(t)v - Delta v = vertical bar partial derivative(t)u vertical bar((n+1)/(n-1)) + vertical bar u vertical bar(q) in n (>= 2) space dimensions. We show the existence of a certain "critical curve" in the pq-plane such that for any (p,q) (p,q > 1) lying below the curve, nonexistence of global solutions occurs, whereas for any (p, q) (p > 1 + 3/(n - 1), q > 1 + 2/(n - 1)) lying exactly on it, this system admits a unique global solution for small data. When n = 3, the discussion for the above system with (p, q) = (3, 3), which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of n = 2 and p = 4 it is observed that no matter how large q is, the higher-order term vertical bar u vertical bar(q) never becomes negligible and it essentially affects the lifespan of small solutions.