具有边界阻尼和非线性对数源项的波方程的能量衰减

Energy decay of wave equation with boundary damping termand nonlinear logarithmic source term

对数型源项问题广泛存在于膨胀宇宙学、超对称场论和量子力学中.为了深入研究对数型源项对波方程能量衰减的影响,讨论了具有边界阻尼、线性强阻尼、非线性弱阻尼和非线性源项的波方程的初边值问题.在合适的假设条件下,通过构造稳定集和Lyapunov泛函,利用势阱方法分别证明了在E(0)<d和E(0)=d的情形下能量的指数衰减性质.

The problem of logarithmic source term widely exists in expansive cosmology,supersymmetric field theory and quantum mechanics.In order to further study the influence of logarithmic source term on the energy decay of wave equation,the initial boundary value problem of wave equation with boundary damping,linear strong damping,nonlinear weak damping and nonlinear source term is discussed.By constructing a stable set and a Lyapunov functional under appropriate assumptions,the exponential decay properties of energy in the case of E(0)<d and E(0)=d are proved by potential well method,respectively.