Note on multiplicative perturbation of local \(C\)-regularized cosine functions with nondensely defined generators. (English) Zbl 1217.47083

Li, Fang; Liu, James H.

Electron. J. Qual. Theory Differ. Equ. 2010, Paper No. 57, 12 p. (2010).

Let \(X\) be a Banach space, \(A\) a linear operator in \(X\). Then it is known that the cosine operator function is the main propagator of the Cauchy problem
\[ u''(t)=Au(t), \quad t\in (-\infty,\infty),\qquad u(0)=0,\quad u'(0)=u_1. \] If \(A\) is the generator of a \(C\)-regularized cosine function \(\{C(t)\}_{t\in\mathbb R}\), then \(u(t)=C^{-1}C(t)u_0+C^{-1}\int^t_0C(s)u_1\,ds\) is the unique solution of the above Cauchy problem for every pair \((u_0,u_1)\) of initial values in \(C(D(A))\), where \(C:=C(0)\).
The authors study the multiplicative perturbation of local \(C\)-regularized cosine functions with nondensely defined generators, in the case when the range of the regularizing operator \(C\) is not dense in \(X\) and the operator \(C\) may not commute with the perturbation operator.
The authors also present an application of the obtained abstract result to the Cauchy problem in \(C_0(\Omega)\) for some integrodifferential equation, and prove that the problem has a unique solution in \(C^2([0,\tau];C_0(\Omega))\), where \(\Omega\) is a domain in \(\mathbb R^n\) and \(C_0(\Omega):=\{f\in C(\Omega)\): for each \(\varepsilon >0\) there is a compact \(\Omega_{\varepsilon}\subset\Omega\) such that \(|f(s)|<\varepsilon\) for all \(s\in\Omega\setminus \Omega_{\varepsilon}\}\), for every couple of initial values in a large subset of \(C_0(\Omega)\).

Reviewer: Yakov Yakubov (Tel-Aviv)

MSC:

47D09	Operator sine and cosine functions and higher-order Cauchy problems
34G10	Linear differential equations in abstract spaces
35G10	Initial value problems for linear higher-order PDEs

Keywords:

multiplicative perturbation; local C-regularized cosine functions; second order differential equation; Cauchy problem; integrodifferential equation