Well-posedness of boundary value problems for a second-order equation
Let \(E\) be a Banach space and let \(A\) be a positive operator acting on \(E\). The authors deduce an estimation for any weak solution of the equation \[ {d^2 u(t) \over d t^2} = A u(t), \quad 0 \leq t \leq T \leq \infty,\tag{1} \] if \(-A\) is the generator of an exponentially stable \(C_0\)-semigroup. Moreover, they establish conditions such that a weak solution is weakened. After that, the conclusions are extended to the study of the representation of bounded solutions for equation (1) and their weakened-type properties.
scientific article; zbMATH DE number 5656110