Optimal lower bound estimates for the blow-up rate for the Zakharov system in a nonhomogeneous medium

In this paper we obtain lower bound estimates for the blow-up rate of finite time blow-up solutions to the Cauchy problem for the Zakharov system in a nonhomogeneous medium in two space dimensions. By introducing suitable scale transformations of space and time, and the use of compactness arguments, we derive an optimal lower bound estimate in the energy space H2(R2)×L2(R2)×H1(R2) for the blow-up rate for t near the finite blow-up time T. Also we give an application to the virial identity for the Zakharov system under study.