一类四阶非线性波动方程解的爆破与衰减

FINITE TIME BLOW UP AND DECAY ESTIMATES OF SOLUTION FOR A CLASS OF FOURTH ORDER NONLINEAR WAVE EQUATION

本文研究了非线性阻尼项与源项的竞争对具有强阻尼项的四阶波动方程解的影响.利用不动点原理和势井方法给出了方程局部弱解存在唯一性满足的条件,证明了当m<p且初始能量E(0)<0时,解将在有限时间内爆破.同时对m,p的大小关系不加任何限制但存在t_0使0<E(t_0)<d的情况下,利用稳定集,研究了整体解的存在性,并得到了解的能量衰减估计.最后借助修正的能量泛函,指出当m≥p时弱解也是整体存在的,推广并改进了文献[1-6]中的结果.

In this paper,we consider the fourth order wave equation with nonlinear damping and source term.We prove the existence of a local weak solution and show that this solution blow up in finite time if m p and the energy is negative.Furthermore,we discuss,for the evolution of solution enters into the stable set,the solution is global as well as a decay result regardless of any relations between m and p.At last we also show that the solution is global if m p,which extends and improves the results in[1-6].