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

TL;DR

This paper investigates the well-posedness and blow-up behavior of solutions to the generalized Zakharov equations with magnetic field in R^d, establishing existence, uniqueness, blow-up rate bounds, and long-term growth characteristics.

Contribution

It proves the well-posedness of the GZ system with magnetic field and derives the lower bound of blow-up rate, along with analyzing the exponential growth of solutions over time.

Findings

Existence and uniqueness of solutions in Sobolev spaces.

Lower bound of blow-up rate near critical index.

Exponential growth of the H^k-norm for global solutions.

Abstract

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.