一类含有非线性对数源项的Kirchhoff 型方程解的爆破

Blow up of solutions for a class of Kirchhoff type equations with nonlinear logarithmic source term

摘要:
在初始能量 E(0) \in (0,E_1) 时，利用能量法证明了如下含有非线性对数源项的Kirchhoff 型方程解的爆破性：

u_tt - M(t)\Delta u + u + \left( g*\Delta u \right)(t) + \left| u_t \right|^ru_t - \Delta u_t + \left| u \right|^2u = u\ln \left| u \right|^k.

当 q > 1,0 < r < 2 时，方程的解在有限时间点处爆破；当 q \geqslant 1,r = 0 时，方程的解在无限时间点处爆破；q,r 取其它值时，方程整体解存在且能量函数具有指数衰减性.

Abstract: In the initial energy E(0) \in (0,E_1), by the energy method the Kirchhoff-type equations with nonlinear logarithmic source term is studied blow-up of the situation for: u_tt - M(t)\Delta u + u + \left( g*\Delta u \right)(t) + \left| u_t \right|^ru_t - \Delta u_t + \left| u \right|^2u = u\ln \left| u \right|^k. It is proved that if q > 1,0 < r < 2, the solution of the equation blow-up at a finite time point; if q \geqslant 1,r = 0, the solution of the equation blow-up at an infinite time point; in q,r other cases, the equation exists global solutions, and the energy function has exponential decay.