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Asymptotic analysis of the abstract telegraph equation. (English) Zbl 1224.35293
Summary: It is known that each solution of the telegraph equation \(u''(t)+2au'(t)+A^2u(t)= 0\), (\(A=A^{*}\) on \(\mathcal {H}\), \(a > 0\)) is approximately equal to some solution of the abstract heat equation \(2av'(t) + A^2v(t) = 0\). It is shown how to find \(v(0)\) in terms of \(u(0)\) and \(u'(0)\), so that one can say that a given solution of the first equation is like the specific solution of the second.

MSC:
35L90 Abstract hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K90 Abstract parabolic equations
34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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