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Semigroups associated with dissipative systems. (English) Zbl 0924.73003

Chapman & Hall/CRC Research Notes in Mathematics. 398. Boca Raton, FL: Chapman & Hall/ CRC. 206 p. (1999).
Linear boundary value problems for the equations of elasticity and viscoelasticity are reduced to an evolution operator equation \(y'=Ay\) in a Hilbert space \(H\), and exponential stability and analyticity of the corresponding semigroup \(S(t)\) are studied on the basis of the following two theorems. First, the semigroup \(S(t)\), which is a semigroup of contractions on \(H\), is exponentially stable if and only if \(iR\subset\rho (A)\) and \(\limsup\| (i\beta I-A)^{-1}\| <\infty\) as \(| \beta | \to \infty\), where \(\rho (A)\) is the resolvent set of \(A\). Second, the semigroup \(S(t)\), which is a semigroup of contractions on \(H\), is analytic if and only if \(\limsup\| \beta (i\beta I-A)^{-1}\| <\infty\), as \(| \beta | \to \infty\), provided \(iR\subset\rho (A)\). More exactly, exponential stability or analyticity is verified by the contradiction argument. Under the assumption that one of the necessary conditions is not true, a sequence \(y_n\), \(\| y_n\| =1\), of solutions of corresponding to elliptic boundary value problems (elastic or viscoelastic) is constructed to arrive at a contradiction due to properties of these boundary value problems.
By this systematic approach, all the considered boundary value problems are shown to exhibit the properties of exponential stability and analyticity, otherwise references are provided for the results of instability and non-analyticity. Heat conduction, viscous damping, and boundary damping are indicated among the dissipative mechanisms. It is pointed out by the due reference that exponential stability does not hold for the three-dimensional equations of thermoelasticity unless some assumptions on domain and initial data are made. Some numerical approximations for the linear thermoelastic and viscoelastic systems are included, with a study of uniform exponential stability.

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74B99 Elastic materials
35L99 Hyperbolic equations and hyperbolic systems
47D06 One-parameter semigroups and linear evolution equations
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
35Q72 Other PDE from mechanics (MSC2000)
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