# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] ?Lebesgue spaces of differentiable functions and distributions?, Proc. Symp. Pure Math., Vol. 5, 1961, pp. 33-49. [2] Partial Differential Equations, Berkeley Mathematics Lecture Notes, Vol. 3a, 1994. [3] Hoff, Ind. Univ. Math. J. 44 pp 603– (1995) · Zbl 0842.35076 [4] Hoff, Z. Angew. Math. Phys. 48 pp 597– (1997) · Zbl 0882.76074 [5] Kawashima, Commun. Math. Phys. 148 pp 189– (1992) · Zbl 0779.35066 [6] Lizorkin, Dokl. Akad. Nauk. SSSR 152 pp 808– (1963) [7] Marcinkiewicz, Studia Math. 8 pp 78– (1939) [8] Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford, 1965. [9] The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973. [10] The Analysis of Linear Partial Differential Operators I, Grund. Math. Wiss., Vol. 256, Springer, Berlin, 1983. [11] and ?A decay property of the Fourier image and its application to Stokes problem and interface problem, preprint in 1999. [12] ?On Dirichlet boundary value problem?, An Lp-Theory Based on a Generalization of Gårding’s Inequality, Lecture Notes in Mathematics, Vol. 268, Springer, Berlin, 1972. [13] Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501 [14] Liu, Commun. Math. Phys. 196 pp 145– (1998) · Zbl 0912.35122 [15] Asymptotic Methods in Equations of Mathematical Physics, Gordon and Beach Science Publishers, New York, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.