高频超声中带退化记忆项的MGT方程解的衰减估计

Decay of Solution for the MGT Equations with Degenerate Memory Arising in High Frequency Ultrasound

研究了高频超声应用中带无穷退化记忆项的Moore-Gibson-Thompson方程τu_(ttt)+α(x)u_(tt)-c^(2)Δu-bΔu_(t)+∫^(∞)_(0)g(s)div[a(x)▽u(t-s)]ds=0解的适定性和衰减速率,其中非负函数a(x)和α(x)是可退化的并满足a(x)+α(x)≥δ>0.该系统是由黏性热松弛流体中波传播模型的线性化而得到的.通过使用Faedo-Galerkin逼近和能量估计,证明了解的适定性.在适当的假设下,通过构造适当的李雅普诺夫泛函,建立了能量的指数或一般衰减结果.

This paper is concerned with a third order in time dissipative Moore-Gibson-Thompson equation with infinite degenerate memoryτu_(ttt)+α(x)u_(tt)-c^(2)Δu-bΔu_(t)+∫^(∞)_(0)g(s)div[a(x)▽u(t-s)]ds=0 in which the function a(x)≥0 andα(x)≥0 can be degenerate but satisfy a(x)+α(x)≥δ>0.This equation arises as a linearization of a model for wave propagation in viscous thermally relaxing fluids.By using the Faedo-Galerkin approximations together with some energy estimates,we prove that the above system is well-posedness.Besides,under appropriate assumptions,we establish the exponential or general decay results of energy via constructing suitable Lyapunov functionals.