## Non-autonomous approximations governed by the fractional powers of damped wave operators.(English)Zbl 1415.35178

Summary: In this article we study non-autonomous approximations governed by the fractional powers of damped wave operators of order $$\alpha \in (0,1)$$ subject to Dirichlet boundary conditions in an $$n$$-dimensional bounded domain with smooth boundary. We give explicitly expressions for the fractional powers of the wave operator, we compute their resolvent operators and their eigenvalues. Moreover, we study the convergence as $$\alpha\nearrow 1$$ with rate $$1-\alpha$$.

### MSC:

 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L05 Wave equation 35B40 Asymptotic behavior of solutions to PDEs 35R11 Fractional partial differential equations
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### References:

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