Integral representation and stabilization of the solution to the Cauchy problem for an equation with two noncommuting operators

A. V. Glushak

Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 38-47

Résumé

We obtain an integral representation for the solution to the Cauchy problem $$ \begin {gathered} \frac{dv}{dt}=\mathbb B_1^2v+\frac 12b(t)(\mathbb B_2\mathbb B_1 +\mathbb B_1\mathbb B_2)v+c(t)\mathbb B_2^2v, \quad v(0)=v_0, \end {gathered} $$ where the operators $\mathbb{B}_1 $ and $\mathbb{B}_2 $ are the infinitesimal generators of strongly continuous groups and $\mathbb B_1\mathbb B_2-\mathbb B_2\mathbb B_1=k\mathbf 1$, $k\ne0$. For the case in which $k=ik_1$, $k_1\in\mathbb R$, it is proved that the solution tends to zero as $t\to+\infty$.