一类具变指数源项的Klein-Gordon方程解的爆破

The Blow-Up of Solutions to the Klein-Gordon Equation with Variable Exponent Source Term

摘要: 本文研究了一类具有变指数源项和非线性阻尼项的Klein-Gordon方程在低初始能量条件下弱解的爆破性质，探讨解在有限时间内爆破的条件及爆破时间的上界估计。我们运用微分不等式技术，通过定义能量泛函和辅助函数，利用Sobolev嵌入不等式、Hölder不等式等工具，分析低初始能量下解的行为，推导解的爆破条件。在给定的变指数条件和松弛函数假设下，证明了当初始能量低于某一临界值时，弱解在有限时间内发生爆破，并获得了爆破时间的上界估计。变指数源项和非线性阻尼项的共同作用会导致解在低初始能量下有限时间爆破，研究结果为该类方程的解的定性分析提供了理论依据，拓展了变指数非线性偏微分方程的爆破理论。

Abstract: This paper studies the blow-up properties of weak solutions of a class of Klein-Gordon equations with variable exponential source term and nonlinear damping term under the condition of low initial energy, and explores the conditions for the blow-up of solutions within a finite time and the upper bound estimation of the blow-up time. We apply differential inequality techniques, and by defining energy functionals and auxiliary functions, and using tools such as Sobolev embedding inequalities and Hölder inequalities, we analyze the behavior of solutions under low initial energy and derive the blow-up conditions of the solutions. Under the given variable exponential condition and the assumption of the relaxation function, it is proved that when the initial energy is lower than a certain critical value, the weak solution blows up within a finite time, and the upper bound estimation of the blow-up time is obtained. The combined effect of the variable exponential source term and the nonlinear damping term can lead to the blow-up of the solution in a finite time at a low initial energy. The research results provide a theoretical basis for the qualitative analysis of the solutions of this type of equation and expand the blow-up theory of variable exponential nonlinear partial differential equations.