Finite-time blowup for Zakharov system with combined power-type nonlinearities

Abstract

This article is concerned with finite-time blowup results for solutions of a class of generalized Zakharov system with combined power-type nonlinearities in three space dimensions:

On the one hand, for c ₀ = +∞ (in the subsonic limit), if α ≥ 0 and or α < 0 and , we obtain that the solution of (GZS) blows up in finite time in the energy space H ¹(R ³). On the other hand, for 0 < c ₀ < +∞, if α ≥ 0, and the solution (ϵ, n, v)(t) of (GZS) are radially symmetric for all t, we prove a new finite-time blowup result of solutions to (GZS) in the energy space H ¹(R ³) × L ²(R ³) × L ²(R ³).

Keywords:

• generalized Zakharov system
• finite-time blowup
• liapunov function in time
• radially symmetric solution

AMS Subject Classifications::

• 35A15
• 35Q55
• 35B30

Acknowledgements

This work is supported by National Natural Science Foundation of P.R.China (Nos 10801102, 10771151).