An abstract version of the fourth-order equation$$\partial_{tttt}u+\alpha\partial_{ttt}u+\beta\partial_{tt}u-\gamma\Delta\partial_{tt}u-\delta\Delta\partial_{t}u-\varrho\Delta u=0$$subject to the homogeneous Dirichlet boundary condition is analyzed. Such a model encompasses the Moore–Gibson–Thompson equation with memory in presence of an exponential kernel. The stability properties of the related solution semigroup are investigated. In particular, a necessary and sufficient condition for exponential stability is established, in terms of the values of certain stability numbers depending on the strictly positive parameters \({\alpha, \beta, \gamma, \delta, \varrho}\).

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关于Moore–Gibson–Thompson型的四阶方程

四阶方程的抽象形式$$ \ partial_ {tttt} u + \ alpha \ partial_ {ttt} u + \ beta \ partial_ {tt} u- \ gamma \ Delta \ partial_ {tt} u- \ delta \ Delta \分析了均质Dirichlet边界条件下的part_ {t} u- \ varrho \ Delta u = 0 $$。这样的模型包含具有指数核的内存的Moore-Gibson-Thompson方程。研究了相关解半群的稳定性。特别地，根据依赖于严格正参数\（{\ alpha，\ beta，\ gamma，\ delta，\ varrho} \）的某些稳定性数的值，建立了指数稳定性的必要和充分条件。