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On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type. (English) Zbl 1269.35008

Summary: In this paper, we study the Riesz basis property of the generalized eigenfunctions of a one-dimensional hyperbolic system in the energy state space. This characterizes the dynamic behavior of the system, particularly the stability, in terms of its eigenfrequencies. This system is derived from a thermoelastic equation with memory type. The asymptotic expansions for eigenvalues and eigenfunctions are developed. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the \(C_0\)-semigroup associated with the system, and as a consequence, the exponential stability of the system is concluded.

MSC:

35J10 Schrödinger operator, Schrödinger equation
93C20 Control/observation systems governed by partial differential equations
93C25 Control/observation systems in abstract spaces
47E05 General theory of ordinary differential operators
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References:

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