In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some $L^2$ estimates of solutions, which improve those in the previous research [51]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted $L^1$ space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on the obtained sharp $L^2$ estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method.

在本文中，我们研究了在耗散情况下线性和半线性Moore-Gibson-Thompson（MGT）方程的柯西问题。关于线性MGT模型，通过利用与傅立叶分析相关的WKB分析，我们得出了一些${\mathrm{大号}}^{\mathrm{2个}}$解决方案的估计，这改善了先前的研究[51]。此外，解的渐近曲线和加权框架中的近似关系${\mathrm{大号}}^{\mathrm{1个}}$空间是派生的。接下来，借助经典能量方法和哈代不等式，我们获得了能量和解本身的奇异极限结果。关于半线性MGT模型，基于获得的锐度${\mathrm{大号}}^{\mathrm{2个}}$估计和构建时间加权Sobolev空间，我们研究具有不同规则性的Sobolev解的整体（时间）存在性。最后，在初始数据的符号假设下，通过使用检验函数方法证明了全局（及时）弱解的不存在。