一类带有弱耗散项的半线性波动方程的Cauchy问题

给出一类带有弱耗散项的线性波动方程的Cauchy问题的解在Sobolev空间中的衰减估计,引入一个同时体现解的能量估计及解的衰减性的函数空间作为迭代的基本空间,同时建立了一个反映基本空间性质的映射,利用整体迭代法和压缩映射原理,在小初值情形下得出其半线性波动方程右端的非线性项F在满足一定条件的情况下,其Cauchy问题解的存在唯一性及解在t→+∞时的衰减性.

The Cauchy problem for a class of semi-linear wave equations with a weak dissipation term

The decay estimates for the solution in the Soblev space to the Cauchy problem for the wave equation with weakly dissipative term is given.A basic space which can exhibit the energy estimates and decay estimates of the solution is introduced,at the same time,a mapping reflecting characteristics of the space is set up.Then,by global iterative method and contracting mapping principle,the existence,uniqueness and decay estimates as t→+∞for the solution to the Cauchy problem for the semi-linear wave equations with weakly dissipative term was proved with small initial data and under some conditions on nonlinear term F.