Asymptotic analysis of the abstract telegraph equation. (English) Zbl 1224.35293

Clarke, Ted; Eckstein, Eugene C.; Goldstein, Jerome A.

Differ. Integral Equ. 21, No. 5-6, 433-442 (2008).

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

Keywords:

abstract telegraph equation; abstract heat equation; solution