Study on the Longtime Behaviour of theViscoelastic Wave Equations with Time Dependent Memory Kernels and Damping Terms

本文考虑一类具时变记忆核及线性阻尼的黏弹性波动方程的初边值问题。首先，通过引入记忆函数将问题进行转化，建立适当的时变记忆空间，并证明了记忆函数所满足的积分不等式。进一步，利用 Feado-Galerkin 方法证明了弱解的存在性，并得到了解的一致有界性。其次，我们证明了解对初值的连续依赖性，从而得到解的唯一性。最后，在外源项为 0 的条件下，通过构造辅助泛函，利用积分不等式，证明了系统能量的指数衰减。相关方法在一般非时变黏弹性波动方程解的长时间行为研究中仍然适用。This paper is concerned with the initial boundary value problem of a viscoelastic wave equations with time dependent memory kernel and linear damping. Firstly, we introduce a memory function and rewrite the problem to a equivalent system. By constructing an appropriate time dependent memory space, we control the memory function by some integral inequalities. Furthermore, the existence of weak solutions is proved by the Feado-Galerkin method, and the uniform boundedness of the solutions is also obtained. Secondly, we prove the continuous dependence of the solutions on the initial values and obtain the uniqueness of the solution. Finally, by constructing an auxiliary functional and using the integral inequality, we prove the exponential decay of the energy under the condition that the external source term is 0. The related methods are also applicable in the study of the long time behavior of solutions of classical viscoelastic wave equations.