Existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction

Abstract

This paper studies the existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction, which describe the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves in a plasma considering the quantum effects. Firstly the existence and uniqueness of the local smooth solutions are obtained by the a priori estimates and the Galerkin method. Secondly, and what is more, by introducing some auxiliary functionals and invariant manifolds, the authors study and derive a sharp threshold for the global existence and blowup of solutions by applying potential well argument and the concavity method. Furthermore, two more specific conditions of how small the initial data are for the solutions to exist globally are concluded by the dilation transformation.