Stabilization of ternary mixtures with frictional dissipation

We analyze the decay properties of the solution semigroup generated by a linear evolution system modeling a mixture of three interacting continua with frictional damping. In particular, we provide conditions for the strong and exponential stability when the dissipation is contributed only by some equation of the system. Moreover, we study situations where the semigroup decays polynomially or admits trajectories with conserved energy, and we find the optimal polynomial decay rate.

[1]  R. Quintanilla,et al.  Exponential decay in a thermoelastic mixture of solids , 2009 .

[2]  Wolfgang Arendt,et al.  Tauberian theorems and stability of one-parameter semigroups , 1988 .

[3]  Yuri Tomilov,et al.  Optimal polynomial decay of functions and operator semigroups , 2009, 0910.0859.

[4]  R. Quintanilla,et al.  SOME QUALITATIVE RESULTS FOR THE LINEAR THEORY OF BINARY MIXTURES OF THERMOELASTIC SOLIDS , 1995 .

[5]  P. M. Naghdi,et al.  A Dynamical theory of interacting continua , 1965 .

[6]  Mario Bunge,et al.  Classical Field Theories , 1967 .

[7]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[8]  R. Quintanilla.  Existence and exponential decay in the linear theory of viscoelastic mixtures , 2005 .

[9]  R. M. Bowen,et al.  Diffusion in Mixtures of Elastic Materials. , 1969 .

[10]  Zhuangyi Liu,et al.  Semigroups Associated with Dissipative Systems , 1999 .

[11]  R. Quintanilla,et al.  Existence and continuous dependence results in the theory of interacting continua , 1994 .

[12]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[13]  Kumbakonam R. Rajagopal,et al.  Mechanics of Mixtures , 1995 .

[14]  A. Green,et al.  A note on mixtures , 1968 .

[15]  Ramón Quintanilla,et al.  Exponential decay in mixtures with localized dissipative term , 2005, Appl. Math. Lett..

[16]  D. S. Drumheller,et al.  Theories of immiscible and structured mixtures , 1983 .

[17]  R. E. Craine,et al.  CONTINUUM THEORIES OF MIXTURES: BASIC THEORY AND HISTORICAL DEVELOPMENT , 1976 .

[18]  J. Prüss.  On the Spectrum of C 0 -Semigroups , 1984 .

[19]  M. Sepúlveda,et al.  Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity , 2007, 0704.1861.

[20]  M. Sepúlveda,et al.  Exponential stability in thermoviscoelastic mixtures of solids , 2009 .