Blowing up of solutions to the Cauchy problem for the generalized Zakharov system with combined power-type nonlinearities

Abstract

This paper deals with blowing up of solutions to the Cauchy problem for a class of generalized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c0 = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α ≥ 0 and \(1 + \tfrac{4} {N} \leqslant p < \tfrac{{N + 2}} {{N - 2}}\) or α < 0 and \(1 < p < 1 + \tfrac{4} {N}\) (N = 2, 3); the other is established under the condition N = 3, \(1 < p < \tfrac{{N + 2}} {{N - 2}}\) and α(p − 3) ≥ 0. On the other hand, for c0 < +∞ and α(p − 3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study.