On the solvability in a weight space of a third-order operator-differential equation with multiple characteristic
In this paper, the authors study the solvability of the equations \[ \left( -\frac{d}{dt} +A\right) \left( \frac{d}{dt}+A\right)^2 u(t) +A_1 \frac{d^2 u(t)}{dt^2} +A_2 \frac{du(t)}{dt} +A_3 u(t) =f(t), \quad t\in {\mathbb R}_+, \] with initial conditions \( u(0)=u'(0)=0 \), where \(f\in L_{2,k}({\mathbb R}_+ ; H)\), \(u\in W^3 _{2,k}({\mathbb R}_+;H)\), \(H\) is a Hilbert space, and \[ L_{2,k}({\mathbb R}_+ ; H) := \left\{ u(\cdot ): \int^{+\infty} _0 \| u(t)\|^2_H e^{-kt}\, dt <+\infty \right\} , \] \[ W^3_{2,k} ({\mathbb R}_+;H):= u(\cdot ): \int^{+\infty}_0 (\| u''' (t)\|^2_H +\| A^3 u(t)\|^2_H)e^{kt}\, dt <+\infty . \] Several sufficient conditions for solvability of the equation are obtained. No detailed proofs are given.
third order operator differential equation
scientific article; zbMATH DE number 6106430