Liu, Zhuangyi; Rao, Bopeng Characterization of polynomial decay rate for the solution of linear evolution equation. (English) Zbl 1100.47036 Z. Angew. Math. Phys. 56, No. 4, 630-644 (2005). The authors estimate the decay rate of solutions of the linear evolution equation\[ x'(t)={\mathcal A} x(t) ,\quad x(0)=x_0, \]where \({\mathcal A}\) generates a bounded \(C_0\)-semigroup on a Hilbert space. As the main result, it states that if the resolvent operator \((i\beta -{\mathcal A})^{-1}\) in the imaginary axis is of polynomial order \(\beta^l\), \(l>0\), then for any integer \(k >0\), there is a constant \(C_k >0\) such that \[ \| x(t)\| \leq C_k \left(\frac{\ln t}{t}\right)^{k/t} (\ln t)\| x_0\| _{D({\mathcal A}^k)}. \]As applications, three examples about the wave and heat equations are given. Reviewer: Lan Nguyen (Bowling Green) Cited in 2 ReviewsCited in 177 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 35B40 Asymptotic behavior of solutions to PDEs 35L05 Wave equation Keywords:semigroup; polynomial decay rate; frequency domain PDFBibTeX XMLCite \textit{Z. Liu} and \textit{B. Rao}, Z. Angew. Math. Phys. 56, No. 4, 630--644 (2005; Zbl 1100.47036) Full Text: DOI Link