Compactness of the difference between the thermoviscoelastic semigroup and its decoupled semigroup (Q5928723)

The paper deals with a system of two linear integro-partial differential equations with initial and Dirichlet boundary conditions. For such a thermo-viscoelastic model the problems of stabilization and controllability are studied by the author [J. Math. Pures Appl., IX Ser. 77, No. 4, 355-386 (1998; Zbl 0922.35018)] and by him and \textit{G. H. Williams} [Electron. J. Differ. Equ. 1998, Paper No. 17, 11 pp. (1998; Zbl 0907.93010)], respectively. The present paper is devoted to the proof of the compactness of the difference between the semigroup generated by the considered system of thermoelasticity type and the semigroup generated by its decoupled system, respectively. The result obtained generalizes the one established by \textit{D. B. Henry}, \textit{A. Perissinito} and \textit{O. Lopes} [Nonlinear Anal., Theory, Methods, Appl. 21, No. 1, 65-75 (1993; Zbl 0837.73010)].