Perturbing the boundary conditions of the generator of a cosine family
The author considers the second order abstract Cauchy problem \[ \begin{gathered} u'' (t) = A_{\Phi} u (t), \;t \in \mathbb R, \\ u (0) = u_0, \;u' (0) = u_1, \end{gathered} \] where the operator \(A _{\Phi }\) is given by \(D(A _{\Phi }):=\{u\in D(A):Lu=\Phi u\}\) and \(A _{\Phi } u=Au\). Here, the operator \((A, D (A))\) is a given closed and densely defined linear operator defined on a Banach space \(X\), whereas \(L:D(A)\rightarrow \partial X\) and \({\Phi }:D(A)\rightarrow \partial X\) are closed linear operators and \(\partial X\) is another Banach space called boundary space. Imposing conditions on \(L\) and \(\Phi\), the author shows that the above second order abstract Cauchy problem is well-posed and thus generates a cosine operator function on the Banach space \(X\). Some examples are given.
scientific article; zbMATH DE number 6103710
scientific article