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The Well-posedness and Blow-up rate of Solution for the Generalized Zakharov equations with Magnetic field in R^d

The present paper is devoted to the study of the well-posedness and the lower bound of blow-up rate to the Cauchy problem of the generalized Zakharov(GZ) equations with magnetic field in R^d. The work of well-posedness of the GZ system bases on the local well-posedness theory in [9]. At first, the existence, uniqueness and continuity of solution to the GZ system with magnetic field in Rd is proved. Next, we establish the lower bound of blow-up rate of blow-up solution in sobolev spaces to the GZ system, which is almost a critical index. Finally, we obtain the long time behavior of global solution,whose H^k-norm grows at k-exponentially in time.