On dissipative wave equations in Hilbert space
The author treats the linear second-order abstract Cauchy problem \[ \ddot u(t)+ B\dot u(t)+ Au(t)= 0,\quad u(0)= u_0,\;\dot u(0)= v_0 \] on a Hilbert space. Here \(A\) is selfadjoint and positive definite, \(B\) is \(m\)- accretive and invertible, and a compatibility condition for \(A\) and \(B\) is imposed. The problem is solved by rewriting the equation as a system \(\dot w(t)= {\mathcal A} w(t)\), with \[ {\mathcal A}:= \begin{pmatrix} 0 & Id\\ -A & - B\end{pmatrix}, \] and by treating this system on a scale of Hilbert spaces. In the appendix a new proof of a perturbation result for \(m\)-accretive operators is given.
linear second-order abstract Cauchy problem
perturbation result for \(m\)-accretive operators
scientific article; zbMATH DE number 635925