×

On the local-energy decay property for the elastic wave equation with the Neumann boundary condition. (English) Zbl 0795.35061

Let \(\Omega\) be an exterior domain in \(\mathbb{R}^ n\) \((n\geq 3)\) with smooth and compact boundary \(\Gamma\). We consider the isotropic elastic wave equation with the Neumann boundary condition \[ \begin{cases} (A(\partial_ x)- \partial_ t^ 2) u(t,x)=0 &\text{in } \mathbb{R}\times\Omega,\\ N(\partial_ x) u(t,x)=0 &\text{on } \mathbb{R}\times\Gamma,\\ u(0,x)= f_ 0(x),\;\partial_ t u(0,x)= f_ 1(x) \quad &\text{on } \Omega. \end{cases}\tag{1} \] Here \(A(\partial_ x)\) and \(N(\partial_ x)\) are of the forms \(A(\partial_ x)u= \mu \Delta u+ (\lambda+\mu) \text{grad(div } u)\), \((N(\partial_ x) u)_ i= \sum_{j=1}^ n \nu_ j (x)\sigma_{ij}(u)|_ \Gamma\) \((i=1,2,\dots,n)\), where \(\nu(x)\) is the unit outer-normal vector to \(\Omega\) at \(x\in\Gamma\). In the above, \(\sigma_{ij}(u)= \lambda(\text{div } u)\delta_{ij}+ 2\mu\varepsilon_{ij}(u)\) is the stress tensor for the isotropic elastic materials, where \(\varepsilon_{ij}(u)= (1/2) (\partial_{x_ i} u_ j+ \partial_{x_ j} u_ i)\) is the strain tensor. We assume that the Lamé constants \(\lambda\) and \(\mu\) are independent of the variables \(t\) and \(x\) and satisfy \(\lambda+ {2\over n}\mu>0\) and \(\mu>0\). For a domain \(D\subset \mathbb{R}^ n\) we define the local energy \({\mathbf E}(u,D,t)\) for the solution of (1) in \(D\cap \Omega\) as \[ {\mathbf E}(u,D,t)= {\textstyle{1\over 2}} \int_{D\cap\Omega} \bigl\{ \lambda| \text{div } u(t,x)|^ 2+\mu \sum_{i,j=1}^ n | \varepsilon_{ij} (u(t,x))|^ 2+ | \partial_ t u(t,x)|^ 2 \bigr\}dx. \] We say that problem (1) has the uniform local-energy decay property of strong type when, for any bounded domains \(D\) and \(D_ 0\), there exists a bounded, continuous, and nonnegative-valued function \(p(t)\) defined on \([0,\infty)\) satisfying \[ \int_ 0^ \infty p(t)^{1/2} dt<\infty \qquad \text{and} \qquad \int_ 0^ \infty \int_ s^ \infty p(t)^{1/2} dt ds<\infty \] such that \({\mathbf E}(u,D,t)\leq p(t) {\mathbf E}(u,\Omega,0)\) for any \(t\geq 0\) holds for any solution of (1) with initial data \(f_ 0, f_ 1\in C_ 0^ \infty (D_ 0\cap \overline{\Omega})\). The main theorem in the present paper is the following one.
Theorem. Problem (1) does not have the uniform local-energy decay property of strong type.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] J. D. Achenbach, Wave Propagation in Elastic Solids , North-Holland, New York, 1973. · Zbl 0268.73005
[2] P. G. Ciarlet, Mathematical elasticity. Vol. I , Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. · Zbl 0648.73014
[3] G. Fichera, Existence theorems in elasticity , Mechanics of Solids II eds. S. Flügge and C. Truesdell, vol. 6a, Springer-Verlag, Berlin, 1965, Handbuch Physik, pp. 347-389.
[4] J.-C. Guillot, Existence and uniqueness of a Rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space , Math. Methods Appl. Sci. 8 (1986), no. 2, 289-310. · Zbl 0606.73024
[5] L. Hörmander, The analysis of linear partial differential operators. III , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. · Zbl 0601.35001
[6] W. Ichinose, The Cauchy problem for Schrödinger type equations with variable coefficients , Osaka J. Math. 24 (1987), no. 4, 853-886. · Zbl 0654.35042
[7] W. Ichinose, On \(L^ 2\) well posedness of the Cauchy problem for Schrödinger type equations on the Riemannian manifold and the Maslov theory , Duke Math. J. 56 (1988), no. 3, 549-588. · Zbl 0713.58055
[8] M. Ikawa, Mixed problems for the wave equation. III. Exponential decay of solutions , Publ. Res. Inst. Math. Sci. 14 (1978), no. 1, 71-110. · Zbl 0385.35039
[9] M. Ikehata and G. Nakamura, Decaying and nondecaying properties of the local energy of an elastic wave outside an obstacle , Japan J. Appl. Math. 6 (1989), no. 1, 83-95. · Zbl 0696.73017
[10] H. Iwashita and Y. Shibata, On the analyticity of spectral functions for some exterior boundary value problems , Glas. Mat. Ser. III 23(43) (1988), no. 2, 291-313. · Zbl 0696.35120
[11] B. V. Kapitonov, On the behaviour as \(t\rightarrow \infty\) of the solutions of the exterior boundary value problem for a hyperbolic system , Siberian Math. J. 28 (1987), 444-457. · Zbl 0645.35059
[12] H. Kumano-Go, Pseudodifferential operators , MIT Press, Cambridge, Mass., 1981. · Zbl 0489.35003
[13] V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics , Mathematical Physics and Applied Mathematics, vol. 7, D. Reidel Publishing Co., Dordrecht, 1981. · Zbl 0458.58001
[14] C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation , Comm. Pure Appl. Math. 14 (1961), 561-568. · Zbl 0101.07701
[15] C. S. Morawetz, Exponential decay of solutions of the wave equation , Comm. Pure Appl. Math. 19 (1966), 439-444. · Zbl 0161.08002
[16] J. A. Nitsche, On Korn’s second inequality , RAIRO Anal. Numér. 15 (1981), no. 3, 237-248. · Zbl 0467.35019
[17] J. Ralston, Note on the decay of acoustic waves , Duke Math. J. 46 (1979), no. 4, 799-804. · Zbl 0427.35043
[18] Y. Shibata and H. Soga, Scattering theory for the elastic wave equation , Publ. Res. Inst. Math. Sci. 25 (1989), no. 6, 861-887. · Zbl 0714.35066
[19] H. Soga, Mixed problems for the wave equation with a singular oblique derivative , Osaka J. Math. 17 (1980), no. 1, 199-232. · Zbl 0429.35045
[20] M. E. Taylor, Rayleigh waves in linear elasticity as a propagation of singularities phenomenon , Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 273-291. · Zbl 0432.73021
[21] B. R. Vainberg, On the short wave asymptotic behavior of solutions of stationary problems and the asymptotic behaviour as \(t\rightarrow \infty\) of non-stationary problems , Russian Math. Surveys 30 (1975), 1-58. · Zbl 0318.35006
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.