常微分不等式及其在半线性波动方程的应用

Improved Ordinary Differential Inequality and Its Application to Semilinear Wave Equations

该文首先得到两类变系数的常微分不等式的爆破结果,可视为文献[3,定理3.1]的推广.其次,作为改进的常微分不等式的一个应用,考虑具有尺度不变阻尼项的半线性波动方程的柯西问题,给予初值合理假设,得到当μ>1和1<p<1+2/n时解的生命跨度的上界估计.该结果的证明方法主要来自于文献[11].

In this paper,we first derive some blow-up results for two ordinary differential inequalities with variable coefficients,which are the generalizations of Theorem 3.1 in Li and Zhou[3].Second,as an application of the improved ordinary differential inequality,we consider the Cauchy problem for the semilinear wave equation with scale-invariant damping and deduce the upper bound of the lifespan for the caseμ>1 and 1<p<1+2/n under some suitable assumptions for the initial data.The method for the latter result is due to Lai and Zhou[11].