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Characterization of polynomial decay rate for the solution of linear evolution equation. (English) Zbl 1100.47036

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.

MSC:

47D06 One-parameter semigroups and linear evolution equations
35B40 Asymptotic behavior of solutions to PDEs
35L05 Wave equation
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