zbMATH — the first resource for mathematics

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.

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