一类带对数阻尼项的类波方程解的整体存在性与爆破

GLOBAL EXISTENCE AND BLOW-UP FOR A CLASS OF WAVE-LIKE EQUATIONS WITH LOG-DAMPING TERMS

该文研究一类具有对数阻尼项的对数型类波方程的柯西问题.通过Fourier变换和Laplace变换得到了相应线性问题的衰减估计.基于此衰减估计,建立合适的求解空间,运用压缩映像原理得到了该问题在小初值条件下存在整体解,利用测试函数法证明解在有限时刻爆破.推广了带结构阻尼项波动方程的有关结论.

The purpose of this paper is to study the Cauchy problem of a class of logarithmic wave like equations with logarithmic damping mechanism.The decay estimation of the corresponding linear problem was obtained through Fourier transform and Laplace transform.Based on this decay estimation,a suitable solution space was established,and the contraction mapping principle was applied to obtain the global solution of the problem under small initial conditions.The test function method was used to prove that the solution explodes at a finite time.Some conclusions about wave equation with structural damping term are extended.