Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations
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Abstract
We consider second kind integral equations of the form x(s)  ʃ_{Ω} k(s, t)x(t) dt = y(s) (abbreviated x  Kx = y), in which Ω is some unbounded subset of R^{n} Let X_{p} denote the weighted space of functions x continuous on Ω and satisfying x(s) = O(s ^{p}), s → ∞. We show that if the kernel k(s, t) decays like  s  t  ^{q} as  s  t  → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∊ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I  K)^{1} ∊ B(X_{0}), then (I  K)^{1} ∊ B(X_{p}) for 0 ≤ p < q, and (I  K)^{1} ∊ B(X_{q}) if further conditions on k hold. Thus, if k(s, t) = 0( s  t  ^{q}),  s  t  → ∞, and y(s) = O( s  ^{p}), s → ∞, the asymptotic behavior of the solution x may be estimated as x(s) = O( s  ^{r}),  s  → ∞, r:= min(p, q). The case when k(s, t) = k(s  t), so that the equation is of WienerHopf type, receives especial attention. Conditions, in terms of the symbol of I  K, for I  K to be invertible or Fredholm on Xp are established for certain cases (Ω a halfspace or cone). A boundary integral equation, which models threedimensional acoustic propagation above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation models, in particular, road traffic noise propagation along an infinite road surface surrounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground.
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External organisations 

Research areas and keywords  Subject classification (UKÄ) – MANDATORY

Original language  English 

Pages (fromto)  303327 
Number of pages  25 
Journal  Journal of Integral Equations and Applications 
Volume  7 
Issue number  3 
Publication status  Published  1995 
Publication category  Research 
Peerreviewed  Yes 
Externally published  Yes 