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Keywords:
Banach space; differential equation; linear nonautonomous equation; exponential stability; commutator; parabolic equation
Banach space; differential equation; linear nonautonomous equation; exponential stability; commutator; parabolic equation
Summary:
We consider the equation ${\rm d}y(t)/{\rm d}t=(A+B(t))y(t)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup $({\rm e}^{At})_{t\ge 0}$ on a Banach space ${\cal X}$, $B(t)$ is a variable bounded operator in ${\cal X}$. It is assumed that the commutator $K(t)=AB(t)-B(t)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\|S {\rm e}^{At}\|$ is integrable and the operator $K(t)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations.
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