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On the rate of decay of solutions to linear viscoelastic equation. (English) Zbl 0947.35020
The author studies decay rates of solutions to the Cauchy problem for the equation of linear viscoelasticity in \(\mathbb{R}^n\): \(v_{tt} -\Delta v -\Delta v_t =0\). By using the Fourier analysis, the Marcinkiewicz multiplier theorem and careful estimating low- and high-frequency parts of the solutions, the author obtains the \(L^p-L^q\)-decay estimates (\(1\leq p\leq 2\leq q\leq\infty\)) of the solutions. The dominant asymptotic behavior is given by the convolution of Green functions of the diffusion equation and the wave equation. This paper improves some decay estimates in [D. Hoff and K. Zumbrum, Z. Angew. Math. Phys. 48, 597-614 (1997; Zbl 0882.76074)].
Reviewer: S.Jiang (Beijing)

MSC:
35B40 Asymptotic behavior of solutions to PDEs
74D05 Linear constitutive equations for materials with memory
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients
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