Local energy decay for the wave equation on the exterior of two balls or convex bodies |
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Authors: | De-Fu Liu Nicholas D Kazarinoff |
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Institution: | (1) Department of Applied Mathematics, Beijing Polytechnic University, Beijing;(2) Department of Mathematics, State University of New York at Buffalo, New York |
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Abstract: | An algebraic rate of decay of local energy, nonuniform with respect to the initial data, is established for solutions of the Dirichlet and Neumann problems for the scalar wave equation defined on the exterior V3 of two balls or of two convex bodies. That is, for given initial data f(x)=u(x), 0 and g(x)= u
t
(x, 0), if u solves u
tt
in V with either u(x, t)=0 or u
n
(x,t)+(x) u(x,t,)-0 ((x)0) on V, then there exists a constant T
0, depending upon (f, g), such that the local energy (the energy in any compact set) of u at t=T is bounded from above by QE(0)T
–1 for TT
0, where E(0) is the total initial energy of u and Q is a positive constant, independent of u, that depends upon V. |
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Keywords: | |
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